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### 1 Cartesian Categories

#### 1.1 Cartesian Categories

A 6-tuple ( \mathbf{C}, \times, 1, \alpha, \lambda, \rho ) consisting of

• a category \mathbf{C},

• a functor \times: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C},

• an object 1 \in \mathbf{C},

• a natural isomorphism \alpha_{a,b,c}: a \times (b \times c) \cong (a \times b) \times c,

• a natural isomorphism \lambda_{a}: 1 \times a \cong a,

• a natural isomorphism \rho_{a}: a \times 1 \cong a,

is called a cartesian category, if

• for all objects a,b,c,d, the pentagon identity holds:

(\alpha_{a,b,c} \times \mathrm{id}_d) \circ \alpha_{a,b \times c, d} \circ ( \mathrm{id}_a \times \alpha_{b,c,d} ) = \alpha_{a \times b, c, d} \circ \alpha_{a,b,c \times d},

• for all objects a,c, the triangle identity holds:

( \rho_a \times \mathrm{id}_c ) \circ \alpha_{a,1,c} = \mathrm{id}_a \times \lambda_c.

The corresponding GAP property is given by IsCartesianCategory.

##### 1.1-1 CartesianBraiding
 ‣ CartesianBraiding( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}( a \times b, b \times a ).

The arguments are two objects a,b. The output is the braiding B_{a,b}: a \times b \rightarrow b \times a.

##### 1.1-2 CartesianBraidingWithGivenDirectProducts
 ‣ CartesianBraidingWithGivenDirectProducts( s, a, b, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( a \times b, b \times a ).

The arguments are an object s = a \times b, two objects a,b, and an object r = b \times a. The output is the braiding B_{a,b}: a \times b \rightarrow b \times a.

##### 1.1-3 CartesianBraidingInverse
 ‣ CartesianBraidingInverse( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}( b \times a, a \times b ).

The arguments are two objects a,b. The output is the inverse braiding B_{a,b}^{-1}: b \times a \rightarrow a \times b.

##### 1.1-4 CartesianBraidingInverseWithGivenDirectProducts
 ‣ CartesianBraidingInverseWithGivenDirectProducts( s, a, b, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( b \times a, a \times b ).

The arguments are an object s = b \times a, two objects a,b, and an object r = a \times b. The output is the inverse braiding B_{a,b}^{-1}: b \times a \rightarrow a \times b.

##### 1.1-5 DirectProductOnMorphisms
 ‣ DirectProductOnMorphisms( alpha, beta ) ( operation )

Returns: a morphism in \mathrm{Hom}(a \times b, a' \times b')

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the direct product \alpha \times \beta.

##### 1.1-6 DirectProductOnMorphismsWithGivenDirectProducts
 ‣ DirectProductOnMorphismsWithGivenDirectProducts( s, alpha, beta, r ) ( operation )

Returns: a morphism in \mathrm{Hom}(a \times b, a' \times b')

The arguments are an object s = a \times b, two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = a' \times b'. The output is the direct product \alpha \times \beta.

##### 1.1-7 CartesianAssociatorRightToLeft
 ‣ CartesianAssociatorRightToLeft( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( a \times (b \times c), (a \times b) \times c ).

The arguments are three objects a,b,c. The output is the associator \alpha_{a,(b,c)}: a \times (b \times c) \rightarrow (a \times b) \times c.

##### 1.1-8 CartesianAssociatorRightToLeftWithGivenDirectProducts
 ‣ CartesianAssociatorRightToLeftWithGivenDirectProducts( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( a \times (b \times c), (a \times b) \times c ).

The arguments are an object s = a \times (b \times c), three objects a,b,c, and an object r = (a \times b) \times c. The output is the associator \alpha_{a,(b,c)}: a \times (b \times c) \rightarrow (a \times b) \times c.

##### 1.1-9 CartesianAssociatorLeftToRight
 ‣ CartesianAssociatorLeftToRight( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( (a \times b) \times c \rightarrow a \times (b \times c) ).

The arguments are three objects a,b,c. The output is the associator \alpha_{(a,b),c}: (a \times b) \times c \rightarrow a \times (b \times c).

##### 1.1-10 CartesianAssociatorLeftToRightWithGivenDirectProducts
 ‣ CartesianAssociatorLeftToRightWithGivenDirectProducts( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( (a \times b) \times c \rightarrow a \times (b \times c) ).

The arguments are an object s = (a \times b) \times c, three objects a,b,c, and an object r = a \times (b \times c). The output is the associator \alpha_{(a,b),c}: (a \times b) \times c \rightarrow a \times (b \times c).

##### 1.1-11 CartesianLeftUnitor
 ‣ CartesianLeftUnitor( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(1 \times a, a)

The argument is an object a. The output is the left unitor \lambda_a: 1 \times a \rightarrow a.

##### 1.1-12 CartesianLeftUnitorWithGivenDirectProduct
 ‣ CartesianLeftUnitorWithGivenDirectProduct( a, s ) ( operation )

Returns: a morphism in \mathrm{Hom}(1 \times a, a)

The arguments are an object a and an object s = 1 \times a. The output is the left unitor \lambda_a: 1 \times a \rightarrow a.

##### 1.1-13 CartesianLeftUnitorInverse
 ‣ CartesianLeftUnitorInverse( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(a, 1 \times a)

The argument is an object a. The output is the inverse of the left unitor \lambda_a^{-1}: a \rightarrow 1 \times a.

##### 1.1-14 CartesianLeftUnitorInverseWithGivenDirectProduct
 ‣ CartesianLeftUnitorInverseWithGivenDirectProduct( a, r ) ( operation )

Returns: a morphism in \mathrm{Hom}(a, 1 \times a)

The argument is an object a and an object r = 1 \times a. The output is the inverse of the left unitor \lambda_a^{-1}: a \rightarrow 1 \times a.

##### 1.1-15 CartesianRightUnitor
 ‣ CartesianRightUnitor( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(a \times 1, a)

The argument is an object a. The output is the right unitor \rho_a: a \times 1 \rightarrow a.

##### 1.1-16 CartesianRightUnitorWithGivenDirectProduct
 ‣ CartesianRightUnitorWithGivenDirectProduct( a, s ) ( operation )

Returns: a morphism in \mathrm{Hom}(a \times 1, a)

The arguments are an object a and an object s = a \times 1. The output is the right unitor \rho_a: a \times 1 \rightarrow a.

##### 1.1-17 CartesianRightUnitorInverse
 ‣ CartesianRightUnitorInverse( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(a, a \times 1)

The argument is an object a. The output is the inverse of the right unitor \rho_a^{-1}: a \rightarrow a \times 1.

##### 1.1-18 CartesianRightUnitorInverseWithGivenDirectProduct
 ‣ CartesianRightUnitorInverseWithGivenDirectProduct( a, r ) ( operation )

Returns: a morphism in \mathrm{Hom}(a, a \times 1)

The arguments are an object a and an object r = a \times 1. The output is the inverse of the right unitor \rho_a^{-1}: a \rightarrow a \times 1.

##### 1.1-19 CartesianDiagonal
 ‣ CartesianDiagonal( a, n ) ( operation )

Returns: a morphism in \mathrm{Hom}(a, a^{\times n}).

The arguments are an object a and an integer n \geq 0. The output is the diagonal morphism from a to the n-fold cartesian power a^{\times n}. If the category does not support empty limits, n must be not be 0.

##### 1.1-20 CartesianDiagonalWithGivenCartesianPower
 ‣ CartesianDiagonalWithGivenCartesianPower( a, n, b ) ( operation )

Returns: a morphism in \mathrm{Hom}(a, b)

The arguments are an object a, an integer n, and an object b equal to the n-fold cartesian power a^{\times n} of a. The output is the diagonal morphism from a to b.

##### 1.1-21 LeftCartesianDistributivityExpanding
 ‣ LeftCartesianDistributivityExpanding( a, L ) ( operation )

Returns: a morphism in \mathrm{Hom}( a \times (b_1 \sqcup \dots \sqcup b_n), (a \times b_1) \sqcup \dots \sqcup (a \times b_n) )

The arguments are an object a and a list of objects L = (b_1, \dots, b_n). The output is the left distributivity morphism a \times (b_1 \sqcup \dots \sqcup b_n) \rightarrow (a \times b_1) \sqcup \dots \sqcup (a \times b_n).

##### 1.1-22 LeftCartesianDistributivityExpandingWithGivenObjects
 ‣ LeftCartesianDistributivityExpandingWithGivenObjects( s, a, L, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = a \times (b_1 \sqcup \dots \sqcup b_n), an object a, a list of objects L = (b_1, \dots, b_n), and an object r = (a \times b_1) \sqcup \dots \sqcup (a \times b_n). The output is the left distributivity morphism s \rightarrow r.

##### 1.1-23 LeftCartesianDistributivityFactoring
 ‣ LeftCartesianDistributivityFactoring( a, L ) ( operation )

Returns: a morphism in \mathrm{Hom}( (a \times b_1) \sqcup \dots \sqcup (a \times b_n), a \times (b_1 \sqcup \dots \sqcup b_n) )

The arguments are an object a and a list of objects L = (b_1, \dots, b_n). The output is the left distributivity morphism (a \times b_1) \sqcup \dots \sqcup (a \times b_n) \rightarrow a \times (b_1 \sqcup \dots \sqcup b_n).

##### 1.1-24 LeftCartesianDistributivityFactoringWithGivenObjects
 ‣ LeftCartesianDistributivityFactoringWithGivenObjects( s, a, L, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (a \times b_1) \sqcup \dots \sqcup (a \times b_n), an object a, a list of objects L = (b_1, \dots, b_n), and an object r = a \times (b_1 \sqcup \dots \sqcup b_n). The output is the left distributivity morphism s \rightarrow r.

##### 1.1-25 RightCartesianDistributivityExpanding
 ‣ RightCartesianDistributivityExpanding( L, a ) ( operation )

Returns: a morphism in \mathrm{Hom}( (b_1 \sqcup \dots \sqcup b_n) \times a, (b_1 \times a) \sqcup \dots \sqcup (b_n \times a) )

The arguments are a list of objects L = (b_1, \dots, b_n) and an object a. The output is the right distributivity morphism (b_1 \sqcup \dots \sqcup b_n) \times a \rightarrow (b_1 \times a) \sqcup \dots \sqcup (b_n \times a).

##### 1.1-26 RightCartesianDistributivityExpandingWithGivenObjects
 ‣ RightCartesianDistributivityExpandingWithGivenObjects( s, L, a, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (b_1 \sqcup \dots \sqcup b_n) \times a, a list of objects L = (b_1, \dots, b_n), an object a, and an object r = (b_1 \times a) \sqcup \dots \sqcup (b_n \times a). The output is the right distributivity morphism s \rightarrow r.

##### 1.1-27 RightCartesianDistributivityFactoring
 ‣ RightCartesianDistributivityFactoring( L, a ) ( operation )

Returns: a morphism in \mathrm{Hom}( (b_1 \times a) \sqcup \dots \sqcup (b_n \times a), (b_1 \sqcup \dots \sqcup b_n) \times a)

The arguments are a list of objects L = (b_1, \dots, b_n) and an object a. The output is the right distributivity morphism (b_1 \times a) \sqcup \dots \sqcup (b_n \times a) \rightarrow (b_1 \sqcup \dots \sqcup b_n) \times a .

##### 1.1-28 RightCartesianDistributivityFactoringWithGivenObjects
 ‣ RightCartesianDistributivityFactoringWithGivenObjects( s, L, a, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (b_1 \times a) \sqcup \dots \sqcup (b_n \times a), a list of objects L = (b_1, \dots, b_n), an object a, and an object r = (b_1 \sqcup \dots \sqcup b_n) \times a. The output is the right distributivity morphism s \rightarrow r.

#### 1.2 Cartesian Closed Categories

A cartesian category \mathbf{C} which has for each functor - \times b: \mathbf{C} \rightarrow \mathbf{C} a right adjoint (denoted by \mathrm{Exponential}(b,-)) is called a closed cartesian category.

The corresponding GAP property is called IsCartesianClosedCategory.

##### 1.2-1 ExponentialOnObjects
 ‣ ExponentialOnObjects( a, b ) ( operation )

Returns: an object

The arguments are two objects a,b. The output is the exponential object \mathrm{Exponential}(a,b).

##### 1.2-2 ExponentialOnMorphisms
 ‣ ExponentialOnMorphisms( alpha, beta ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Exponential}(a',b), \mathrm{Exponential}(a,b') )

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the exponential morphism \mathrm{Exponential}(\alpha,\beta): \mathrm{Exponential}(a',b) \rightarrow \mathrm{Exponential}(a,b').

##### 1.2-3 ExponentialOnMorphismsWithGivenExponentials
 ‣ ExponentialOnMorphismsWithGivenExponentials( s, alpha, beta, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = \mathrm{Exponential}(a',b), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{Exponential}(a,b'). The output is the exponential morphism \mathrm{Exponential}(\alpha,\beta): \mathrm{Exponential}(a',b) \rightarrow \mathrm{Exponential}(a,b').

##### 1.2-4 CartesianRightEvaluationMorphism
 ‣ CartesianRightEvaluationMorphism( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}(a \times \mathrm{Exponential}(a,b), b ).

The arguments are two objects a, b. The output is the right evaluation morphism \mathrm{ev}_{a,b}:a \times \mathrm{Exponential}(a,b) \rightarrow b, i.e., the counit of the direct product-exponential adjunction.

##### 1.2-5 CartesianRightEvaluationMorphismWithGivenSource
 ‣ CartesianRightEvaluationMorphismWithGivenSource( a, b, s ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = a \times \mathrm{Exponential}(a,b). The output is the right evaluation morphism \mathrm{ev}_{a,b}: a \times \mathrm{Exponential}(a,b) \rightarrow b, i.e., the counit of the direct product-exponential adjunction.

##### 1.2-6 CartesianRightCoevaluationMorphism
 ‣ CartesianRightCoevaluationMorphism( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{Exponential}(a, a \times b) ).

The arguments are two objects a,b. The output is the right coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, a \times b), i.e., the unit of the direct product-exponential adjunction.

##### 1.2-7 CartesianRightCoevaluationMorphismWithGivenRange
 ‣ CartesianRightCoevaluationMorphismWithGivenRange( a, b, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = \mathrm{Exponential}(a, a \times b). The output is the right coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, a \times b), i.e., the unit of the direct product-exponential adjunction.

 ‣ DirectProductToExponentialRightAdjunctMorphism( a, b, f ) ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{Exponential}(a,c) ).

The arguments are two objects a,b and a morphism f: a \times b \rightarrow c. The output is a morphism g: b \rightarrow \mathrm{Exponential}(a,c) corresponding to f under the direct product-exponential adjunction.

 ‣ DirectProductToExponentialRightAdjunctMorphismWithGivenExponential( a, b, f, i ) ( operation )

Returns: a morphism in \mathrm{Hom}( b, i ).

The arguments are two objects a,b, a morphism f: a \times b \rightarrow c and an object i = \mathrm{Exponential}(a,c). The output is a morphism g: b \rightarrow i corresponding to f under the direct product-exponential adjunction.

 ‣ DirectProductToExponentialRightAdjunctionIsomorphism( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( H(a \times b, c), H(b, \mathrm{Exponential}(a,c)) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(a \times b, c) \to H(b, \mathrm{Exponential}(a,c)) in the range category of the homomorphism structure H.

 ‣ DirectProductToExponentialRightAdjunctionIsomorphismWithGivenObjects( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(a \times b, c) and r = H(b, \mathrm{Exponential}(a,c)). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

 ‣ ExponentialToDirectProductRightAdjunctMorphism( a, c, g ) ( operation )

Returns: a morphism in \mathrm{Hom}(a \times b, c).

The arguments are two objects a,c and a morphism g: b \rightarrow \mathrm{Exponential}(a,c). The output is a morphism f: a \times b \rightarrow c corresponding to g under the direct product-exponential adjunction.

 ‣ ExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct( a, c, g, s ) ( operation )

Returns: a morphism in \mathrm{Hom}(s, c).

The arguments are two objects a,c, a morphism g: b \rightarrow \mathrm{Exponential}(a,c) and an object s = a \times b. The output is a morphism f: s \rightarrow c corresponding to g under the direct product-exponential adjunction.

 ‣ ExponentialToDirectProductRightAdjunctionIsomorphism( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( H(b, \mathrm{Exponential}(a,c)), H(a \times b, c) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(b, \mathrm{Exponential}(a,c)) \to H(a \times b, c) in the range category of the homomorphism structure H.

 ‣ ExponentialToDirectProductRightAdjunctionIsomorphismWithGivenObjects( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(b, \mathrm{Exponential}(a,c)) and r = H(a \times b, c). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

##### 1.2-16 CartesianLeftEvaluationMorphism
 ‣ CartesianLeftEvaluationMorphism( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Exponential}(a,b) \times a, b ).

The arguments are two objects a, b. The output is the left evaluation morphism \mathrm{ev}_{a,b}: \mathrm{Exponential}(a,b) \times a \rightarrow b, i.e., the counit of the direct product-exponential adjunction.

##### 1.2-17 CartesianLeftEvaluationMorphismWithGivenSource
 ‣ CartesianLeftEvaluationMorphismWithGivenSource( a, b, s ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = \mathrm{Exponential}(a,b) \times a. The output is the left evaluation morphism \mathrm{ev}_{a,b}: \mathrm{Exponential}(a,b) \times a \rightarrow b, i.e., the counit of the direct product-exponential adjunction.

##### 1.2-18 CartesianLeftCoevaluationMorphism
 ‣ CartesianLeftCoevaluationMorphism( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{Exponential}(a, b \times a) ).

The arguments are two objects a,b. The output is the left coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, b \times a), i.e., the unit of the direct product-exponential adjunction.

##### 1.2-19 CartesianLeftCoevaluationMorphismWithGivenRange
 ‣ CartesianLeftCoevaluationMorphismWithGivenRange( a, b, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = \mathrm{Exponential}(a, b \times a). The output is the left coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{Exponential}(a, b \times a), i.e., the unit of the direct product-exponential adjunction.

 ‣ DirectProductToExponentialLeftAdjunctMorphism( a, b, f ) ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{Exponential}(b,c) ).

The arguments are two objects a,b and a morphism f: a \times b \rightarrow c. The output is a morphism g: a \rightarrow \mathrm{Exponential}(b,c) corresponding to f under the direct product-exponential adjunction.

 ‣ DirectProductToExponentialLeftAdjunctMorphismWithGivenExponential( a, b, f, i ) ( operation )

Returns: a morphism in \mathrm{Hom}( a, i ).

The arguments are two objects a,b, a morphism f: a \times b \rightarrow c and an object i = \mathrm{Exponential}(b,c). The output is a morphism g: a \rightarrow i corresponding to f under the direct product-exponential adjunction.

 ‣ DirectProductToExponentialLeftAdjunctionIsomorphism( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( H(a \times b, c), H(a, \mathrm{Exponential}(b,c)) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(a \times b, c) \to H(a, \mathrm{Exponential}(b,c)) in the range category of the homomorphism structure H.

 ‣ DirectProductToExponentialLeftAdjunctionIsomorphismWithGivenObjects( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(a \times b, c) and r = H(a, \mathrm{Exponential}(b,c)). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

 ‣ ExponentialToDirectProductLeftAdjunctMorphism( b, c, g ) ( operation )

Returns: a morphism in \mathrm{Hom}(a \times b, c).

The arguments are two objects b,c and a morphism g: a \rightarrow \mathrm{Exponential}(b,c). The output is a morphism f: a \times b \rightarrow c corresponding to g under the direct product-exponential adjunction.

 ‣ ExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct( b, c, g, s ) ( operation )

Returns: a morphism in \mathrm{Hom}(s, c).

The arguments are two objects b,c, a morphism g: a \rightarrow \mathrm{Exponential}(b,c) and an object s = a \times b. The output is a morphism f: s \rightarrow c corresponding to g under the direct product-exponential adjunction.

 ‣ ExponentialToDirectProductLeftAdjunctionIsomorphism( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( H(a, \mathrm{Exponential}(b,c)), H(a \times b, c) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(a, \mathrm{Exponential}(b,c)) \to H(a \times b, c) in the range category of the homomorphism structure H.

 ‣ ExponentialToDirectProductLeftAdjunctionIsomorphismWithGivenObjects( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(a, \mathrm{Exponential}(b,c)) and r = H(a \times b, c). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

##### 1.2-28 CartesianPreComposeMorphism
 ‣ CartesianPreComposeMorphism( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c), \mathrm{Exponential}(a,c) ).

The arguments are three objects a,b,c. The output is the precomposition morphism \mathrm{CartesianPreComposeMorphism}_{a,b,c}: \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c) \rightarrow \mathrm{Exponential}(a,c).

##### 1.2-29 CartesianPreComposeMorphismWithGivenObjects
 ‣ CartesianPreComposeMorphismWithGivenObjects( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c), three objects a,b,c, and an object r = \mathrm{Exponential}(a,c). The output is the precomposition morphism \mathrm{CartesianPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{Exponential}(a,b) \times \mathrm{Exponential}(b,c) \rightarrow \mathrm{Exponential}(a,c).

##### 1.2-30 CartesianPostComposeMorphism
 ‣ CartesianPostComposeMorphism( a, b, c ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b), \mathrm{Exponential}(a,c) ).

The arguments are three objects a,b,c. The output is the postcomposition morphism \mathrm{CartesianPostComposeMorphism}_{a,b,c}: \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b) \rightarrow \mathrm{Exponential}(a,c).

##### 1.2-31 CartesianPostComposeMorphismWithGivenObjects
 ‣ CartesianPostComposeMorphismWithGivenObjects( s, a, b, c, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b), three objects a,b,c, and an object r = \mathrm{Exponential}(a,c). The output is the postcomposition morphism \mathrm{CartesianPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{Exponential}(b,c) \times \mathrm{Exponential}(a,b) \rightarrow \mathrm{Exponential}(a,c).

##### 1.2-32 CartesianDualOnObjects
 ‣ CartesianDualOnObjects( a ) ( attribute )

Returns: an object

The argument is an object a. The output is its dual object a^{\vee}.

##### 1.2-33 CartesianDualOnMorphisms
 ‣ CartesianDualOnMorphisms( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( b^{\vee}, a^{\vee} ).

The argument is a morphism \alpha: a \rightarrow b. The output is its dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.2-34 CartesianDualOnMorphismsWithGivenCartesianDuals
 ‣ CartesianDualOnMorphismsWithGivenCartesianDuals( s, alpha, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The argument is an object s = b^{\vee}, a morphism \alpha: a \rightarrow b, and an object r = a^{\vee}. The output is the dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.2-35 CartesianEvaluationForCartesianDual
 ‣ CartesianEvaluationForCartesianDual( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \times a, 1 ).

The argument is an object a. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \times a \rightarrow 1.

##### 1.2-36 CartesianEvaluationForCartesianDualWithGivenDirectProduct
 ‣ CartesianEvaluationForCartesianDualWithGivenDirectProduct( s, a, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \times a, an object a, and an object r = 1. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \times a \rightarrow 1.

##### 1.2-37 MorphismToCartesianBidual
 ‣ MorphismToCartesianBidual( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(a, (a^{\vee})^{\vee}).

The argument is an object a. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.2-38 MorphismToCartesianBidualWithGivenCartesianBidual
 ‣ MorphismToCartesianBidualWithGivenCartesianBidual( a, r ) ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The arguments are an object a, and an object r = (a^{\vee})^{\vee}. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.2-39 DirectProductExponentialCompatibilityMorphism
 ‣ DirectProductExponentialCompatibilityMorphism( list ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b'), \mathrm{Exponential}(a \times b,a' \times b')).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{DirectProductExponentialCompatibilityMorphism}_{a,a',b,b'}: \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b') \rightarrow \mathrm{Exponential}(a \times b,a' \times b').

##### 1.2-40 DirectProductExponentialCompatibilityMorphismWithGivenObjects
 ‣ DirectProductExponentialCompatibilityMorphismWithGivenObjects( s, list, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b') and r = \mathrm{Exponential}(a \times b,a' \times b'). The output is the natural morphism \mathrm{DirectProductExponentialCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{Exponential}(a,a') \times \mathrm{Exponential}(b,b') \rightarrow \mathrm{Exponential}(a \times b,a' \times b').

##### 1.2-41 DirectProductCartesianDualityCompatibilityMorphism
 ‣ DirectProductCartesianDualityCompatibilityMorphism( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \times b^{\vee}, (a \times b)^{\vee} ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{DirectProductCartesianDualityCompatibilityMorphism}: a^{\vee} \times b^{\vee} \rightarrow (a \times b)^{\vee}.

##### 1.2-42 DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects
 ‣ DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects( s, a, b, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \times b^{\vee}, two objects a,b, and an object r = (a \times b)^{\vee}. The output is the natural morphism \mathrm{DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \times b^{\vee} \rightarrow (a \times b)^{\vee}.

##### 1.2-43 MorphismFromDirectProductToExponential
 ‣ MorphismFromDirectProductToExponential( a, b ) ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \times b, \mathrm{Exponential}(a,b) ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromDirectProductToExponential}_{a,b}: a^{\vee} \times b \rightarrow \mathrm{Exponential}(a,b).

##### 1.2-44 MorphismFromDirectProductToExponentialWithGivenObjects
 ‣ MorphismFromDirectProductToExponentialWithGivenObjects( s, a, b, r ) ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \times b, two objects a,b, and an object r = \mathrm{Exponential}(a,b). The output is the natural morphism \mathrm{MorphismFromDirectProductToExponentialWithGivenObjects}_{a,b}: a^{\vee} \times b \rightarrow \mathrm{Exponential}(a,b).

##### 1.2-45 IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject
 ‣ IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(a^{\vee}, \mathrm{Exponential}(a,1)).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject}_{a}: a^{\vee} \rightarrow \mathrm{Exponential}(a,1).

##### 1.2-46 IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject
 ‣ IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{Exponential}(a,1), a^{\vee}).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject}_{a}: \mathrm{Exponential}(a,1) \rightarrow a^{\vee}.

##### 1.2-47 UniversalPropertyOfCartesianDual
 ‣ UniversalPropertyOfCartesianDual( t, a, alpha ) ( operation )

Returns: a morphism in \mathrm{Hom}(t, a^{\vee}).

The arguments are two objects t,a, and a morphism \alpha: t \times a \rightarrow 1. The output is the morphism t \rightarrow a^{\vee} given by the universal property of a^{\vee}.

##### 1.2-48 CartesianLambdaIntroduction
 ‣ CartesianLambdaIntroduction( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( 1, \mathrm{Exponential}(a,b) ).

The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism 1 \rightarrow \mathrm{Exponential}(a,b) under the direct product-exponential adjunction.

##### 1.2-49 CartesianLambdaElimination
 ‣ CartesianLambdaElimination( a, b, alpha ) ( operation )

Returns: a morphism in \mathrm{Hom}(a,b).

The arguments are two objects a,b, and a morphism \alpha: 1 \rightarrow \mathrm{Exponential}(a,b). The output is a morphism a \rightarrow b corresponding to \alpha under the direct product-exponential adjunction.

##### 1.2-50 IsomorphismFromObjectToExponential
 ‣ IsomorphismFromObjectToExponential( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{Exponential}(1,a)).

The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{Exponential}(1,a).

##### 1.2-51 IsomorphismFromObjectToExponentialWithGivenExponential
 ‣ IsomorphismFromObjectToExponentialWithGivenExponential( a, r ) ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The argument is an object a, and an object r = \mathrm{Exponential}(1,a). The output is the natural isomorphism a \rightarrow \mathrm{Exponential}(1,a).

##### 1.2-52 IsomorphismFromExponentialToObject
 ‣ IsomorphismFromExponentialToObject( a ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{Exponential}(1,a),a).

The argument is an object a. The output is the natural isomorphism \mathrm{Exponential}(1,a) \rightarrow a.

##### 1.2-53 IsomorphismFromExponentialToObjectWithGivenExponential
 ‣ IsomorphismFromExponentialToObjectWithGivenExponential( a, s ) ( operation )

Returns: a morphism in \mathrm{Hom}(s,a).

The argument is an object a, and an object s = \mathrm{Exponential}(1,a). The output is the natural isomorphism \mathrm{Exponential}(1,a) \rightarrow a.

#### 1.3 Convenience Methods

##### 1.3-1 Exponential
 ‣ Exponential( a, b ) ( operation )

Returns: a cell

This is a convenience method. The arguments are two cells a,b. The output is the exponential cell. If a,b are two CAP objects the output is the internal Hom object \mathrm{Exponential}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the exponential on morphisms, where any object is replaced by its identity morphism.

 ‣ AddCartesianBraiding( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianBraiding. F: ( a, b ) \mapsto \mathtt{CartesianBraiding}(a, b).

 ‣ AddCartesianBraidingInverse( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianBraidingInverse. F: ( a, b ) \mapsto \mathtt{CartesianBraidingInverse}(a, b).

 ‣ AddCartesianBraidingInverseWithGivenDirectProducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianBraidingInverseWithGivenDirectProducts. F: ( s, a, b, r ) \mapsto \mathtt{CartesianBraidingInverseWithGivenDirectProducts}(s, a, b, r).

 ‣ AddCartesianBraidingWithGivenDirectProducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianBraidingWithGivenDirectProducts. F: ( s, a, b, r ) \mapsto \mathtt{CartesianBraidingWithGivenDirectProducts}(s, a, b, r).

 ‣ AddCartesianAssociatorLeftToRight( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianAssociatorLeftToRight. F: ( a, b, c ) \mapsto \mathtt{CartesianAssociatorLeftToRight}(a, b, c).

 ‣ AddCartesianAssociatorLeftToRightWithGivenDirectProducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianAssociatorLeftToRightWithGivenDirectProducts. F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianAssociatorLeftToRightWithGivenDirectProducts}(s, a, b, c, r).

 ‣ AddCartesianAssociatorRightToLeft( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianAssociatorRightToLeft. F: ( a, b, c ) \mapsto \mathtt{CartesianAssociatorRightToLeft}(a, b, c).

 ‣ AddCartesianAssociatorRightToLeftWithGivenDirectProducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianAssociatorRightToLeftWithGivenDirectProducts. F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianAssociatorRightToLeftWithGivenDirectProducts}(s, a, b, c, r).

 ‣ AddCartesianDiagonal( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianDiagonal. F: ( a, n ) \mapsto \mathtt{CartesianDiagonal}(a, n).

 ‣ AddCartesianDiagonalWithGivenCartesianPower( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianDiagonalWithGivenCartesianPower. F: ( a, n, cartesian_power ) \mapsto \mathtt{CartesianDiagonalWithGivenCartesianPower}(a, n, cartesian_power).

 ‣ AddCartesianLeftUnitor( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftUnitor. F: ( a ) \mapsto \mathtt{CartesianLeftUnitor}(a).

 ‣ AddCartesianLeftUnitorInverse( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftUnitorInverse. F: ( a ) \mapsto \mathtt{CartesianLeftUnitorInverse}(a).

 ‣ AddCartesianLeftUnitorInverseWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftUnitorInverseWithGivenDirectProduct. F: ( a, r ) \mapsto \mathtt{CartesianLeftUnitorInverseWithGivenDirectProduct}(a, r).

 ‣ AddCartesianLeftUnitorWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftUnitorWithGivenDirectProduct. F: ( a, s ) \mapsto \mathtt{CartesianLeftUnitorWithGivenDirectProduct}(a, s).

 ‣ AddCartesianRightUnitor( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightUnitor. F: ( a ) \mapsto \mathtt{CartesianRightUnitor}(a).

 ‣ AddCartesianRightUnitorInverse( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightUnitorInverse. F: ( a ) \mapsto \mathtt{CartesianRightUnitorInverse}(a).

 ‣ AddCartesianRightUnitorInverseWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightUnitorInverseWithGivenDirectProduct. F: ( a, r ) \mapsto \mathtt{CartesianRightUnitorInverseWithGivenDirectProduct}(a, r).

 ‣ AddCartesianRightUnitorWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightUnitorWithGivenDirectProduct. F: ( a, s ) \mapsto \mathtt{CartesianRightUnitorWithGivenDirectProduct}(a, s).

 ‣ AddDirectProductOnMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductOnMorphisms. F: ( alpha, beta ) \mapsto \mathtt{DirectProductOnMorphisms}(alpha, beta).

 ‣ AddDirectProductOnMorphismsWithGivenDirectProducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductOnMorphismsWithGivenDirectProducts. F: ( s, alpha, beta, r ) \mapsto \mathtt{DirectProductOnMorphismsWithGivenDirectProducts}(s, alpha, beta, r).

 ‣ AddCartesianDualOnMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianDualOnMorphisms. F: ( alpha ) \mapsto \mathtt{CartesianDualOnMorphisms}(alpha).

 ‣ AddCartesianDualOnMorphismsWithGivenCartesianDuals( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianDualOnMorphismsWithGivenCartesianDuals. F: ( s, alpha, r ) \mapsto \mathtt{CartesianDualOnMorphismsWithGivenCartesianDuals}(s, alpha, r).

 ‣ AddCartesianDualOnObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianDualOnObjects. F: ( a ) \mapsto \mathtt{CartesianDualOnObjects}(a).

 ‣ AddCartesianEvaluationForCartesianDual( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianEvaluationForCartesianDual. F: ( a ) \mapsto \mathtt{CartesianEvaluationForCartesianDual}(a).

 ‣ AddCartesianEvaluationForCartesianDualWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianEvaluationForCartesianDualWithGivenDirectProduct. F: ( s, a, r ) \mapsto \mathtt{CartesianEvaluationForCartesianDualWithGivenDirectProduct}(s, a, r).

 ‣ AddCartesianLambdaElimination( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLambdaElimination. F: ( a, b, alpha ) \mapsto \mathtt{CartesianLambdaElimination}(a, b, alpha).

 ‣ AddCartesianLambdaIntroduction( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLambdaIntroduction. F: ( alpha ) \mapsto \mathtt{CartesianLambdaIntroduction}(alpha).

 ‣ AddCartesianLeftCoevaluationMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{CartesianLeftCoevaluationMorphism}(a, b).

 ‣ AddCartesianLeftCoevaluationMorphismWithGivenRange( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftCoevaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{CartesianLeftCoevaluationMorphismWithGivenRange}(a, b, r).

 ‣ AddCartesianLeftEvaluationMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{CartesianLeftEvaluationMorphism}(a, b).

 ‣ AddCartesianLeftEvaluationMorphismWithGivenSource( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianLeftEvaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{CartesianLeftEvaluationMorphismWithGivenSource}(a, b, s).

 ‣ AddCartesianPostComposeMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianPostComposeMorphism. F: ( a, b, c ) \mapsto \mathtt{CartesianPostComposeMorphism}(a, b, c).

 ‣ AddCartesianPostComposeMorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianPostComposeMorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianPostComposeMorphismWithGivenObjects}(s, a, b, c, r).

 ‣ AddCartesianPreComposeMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianPreComposeMorphism. F: ( a, b, c ) \mapsto \mathtt{CartesianPreComposeMorphism}(a, b, c).

 ‣ AddCartesianPreComposeMorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianPreComposeMorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{CartesianPreComposeMorphismWithGivenObjects}(s, a, b, c, r).

 ‣ AddCartesianRightCoevaluationMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightCoevaluationMorphism. F: ( a, b ) \mapsto \mathtt{CartesianRightCoevaluationMorphism}(a, b).

 ‣ AddCartesianRightCoevaluationMorphismWithGivenRange( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightCoevaluationMorphismWithGivenRange. F: ( a, b, r ) \mapsto \mathtt{CartesianRightCoevaluationMorphismWithGivenRange}(a, b, r).

 ‣ AddCartesianRightEvaluationMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightEvaluationMorphism. F: ( a, b ) \mapsto \mathtt{CartesianRightEvaluationMorphism}(a, b).

 ‣ AddCartesianRightEvaluationMorphismWithGivenSource( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation CartesianRightEvaluationMorphismWithGivenSource. F: ( a, b, s ) \mapsto \mathtt{CartesianRightEvaluationMorphismWithGivenSource}(a, b, s).

 ‣ AddDirectProductCartesianDualityCompatibilityMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductCartesianDualityCompatibilityMorphism. F: ( a, b ) \mapsto \mathtt{DirectProductCartesianDualityCompatibilityMorphism}(a, b).

 ‣ AddDirectProductCartesianDualityCompatibilityMorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects. F: ( s, a, b, r ) \mapsto \mathtt{DirectProductCartesianDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r).

 ‣ AddDirectProductExponentialCompatibilityMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductExponentialCompatibilityMorphism. F: ( list ) \mapsto \mathtt{DirectProductExponentialCompatibilityMorphism}(list).

 ‣ AddDirectProductExponentialCompatibilityMorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductExponentialCompatibilityMorphismWithGivenObjects. F: ( source, list, range ) \mapsto \mathtt{DirectProductExponentialCompatibilityMorphismWithGivenObjects}(source, list, range).

 ‣ AddDirectProductToExponentialLeftAdjunctMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialLeftAdjunctMorphism. F: ( a, b, f ) \mapsto \mathtt{DirectProductToExponentialLeftAdjunctMorphism}(a, b, f).

 ‣ AddDirectProductToExponentialLeftAdjunctMorphismWithGivenExponential( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialLeftAdjunctMorphismWithGivenExponential. F: ( a, b, f, i ) \mapsto \mathtt{DirectProductToExponentialLeftAdjunctMorphismWithGivenExponential}(a, b, f, i).

 ‣ AddDirectProductToExponentialLeftAdjunctionIsomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialLeftAdjunctionIsomorphism. F: ( a, b, c ) \mapsto \mathtt{DirectProductToExponentialLeftAdjunctionIsomorphism}(a, b, c).

 ‣ AddDirectProductToExponentialLeftAdjunctionIsomorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialLeftAdjunctionIsomorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{DirectProductToExponentialLeftAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

 ‣ AddDirectProductToExponentialRightAdjunctMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialRightAdjunctMorphism. F: ( a, b, f ) \mapsto \mathtt{DirectProductToExponentialRightAdjunctMorphism}(a, b, f).

 ‣ AddDirectProductToExponentialRightAdjunctMorphismWithGivenExponential( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialRightAdjunctMorphismWithGivenExponential. F: ( a, b, f, i ) \mapsto \mathtt{DirectProductToExponentialRightAdjunctMorphismWithGivenExponential}(a, b, f, i).

 ‣ AddDirectProductToExponentialRightAdjunctionIsomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialRightAdjunctionIsomorphism. F: ( a, b, c ) \mapsto \mathtt{DirectProductToExponentialRightAdjunctionIsomorphism}(a, b, c).

 ‣ AddDirectProductToExponentialRightAdjunctionIsomorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectProductToExponentialRightAdjunctionIsomorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{DirectProductToExponentialRightAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

 ‣ AddExponentialOnMorphisms( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialOnMorphisms. F: ( alpha, beta ) \mapsto \mathtt{ExponentialOnMorphisms}(alpha, beta).

 ‣ AddExponentialOnMorphismsWithGivenExponentials( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialOnMorphismsWithGivenExponentials. F: ( s, alpha, beta, r ) \mapsto \mathtt{ExponentialOnMorphismsWithGivenExponentials}(s, alpha, beta, r).

 ‣ AddExponentialOnObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialOnObjects. F: ( a, b ) \mapsto \mathtt{ExponentialOnObjects}(a, b).

 ‣ AddExponentialToDirectProductLeftAdjunctMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductLeftAdjunctMorphism. F: ( b, c, g ) \mapsto \mathtt{ExponentialToDirectProductLeftAdjunctMorphism}(b, c, g).

 ‣ AddExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct. F: ( b, c, g, s ) \mapsto \mathtt{ExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct}(b, c, g, s).

 ‣ AddExponentialToDirectProductLeftAdjunctionIsomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductLeftAdjunctionIsomorphism. F: ( a, b, c ) \mapsto \mathtt{ExponentialToDirectProductLeftAdjunctionIsomorphism}(a, b, c).

 ‣ AddExponentialToDirectProductLeftAdjunctionIsomorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductLeftAdjunctionIsomorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{ExponentialToDirectProductLeftAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

 ‣ AddExponentialToDirectProductRightAdjunctMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductRightAdjunctMorphism. F: ( a, c, g ) \mapsto \mathtt{ExponentialToDirectProductRightAdjunctMorphism}(a, c, g).

 ‣ AddExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct. F: ( a, c, g, s ) \mapsto \mathtt{ExponentialToDirectProductRightAdjunctMorphismWithGivenDirectProduct}(a, c, g, s).

 ‣ AddExponentialToDirectProductRightAdjunctionIsomorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductRightAdjunctionIsomorphism. F: ( a, b, c ) \mapsto \mathtt{ExponentialToDirectProductRightAdjunctionIsomorphism}(a, b, c).

 ‣ AddExponentialToDirectProductRightAdjunctionIsomorphismWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ExponentialToDirectProductRightAdjunctionIsomorphismWithGivenObjects. F: ( s, a, b, c, r ) \mapsto \mathtt{ExponentialToDirectProductRightAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

 ‣ AddIsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject. F: ( a ) \mapsto \mathtt{IsomorphismFromCartesianDualObjectToExponentialIntoTerminalObject}(a).

 ‣ AddIsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject. F: ( a ) \mapsto \mathtt{IsomorphismFromExponentialIntoTerminalObjectToCartesianDualObject}(a).

 ‣ AddIsomorphismFromExponentialToObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromExponentialToObject. F: ( a ) \mapsto \mathtt{IsomorphismFromExponentialToObject}(a).

 ‣ AddIsomorphismFromExponentialToObjectWithGivenExponential( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromExponentialToObjectWithGivenExponential. F: ( a, s ) \mapsto \mathtt{IsomorphismFromExponentialToObjectWithGivenExponential}(a, s).

 ‣ AddIsomorphismFromObjectToExponential( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromObjectToExponential. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToExponential}(a).

 ‣ AddIsomorphismFromObjectToExponentialWithGivenExponential( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation IsomorphismFromObjectToExponentialWithGivenExponential. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToExponentialWithGivenExponential}(a, r).

 ‣ AddMorphismFromDirectProductToExponential( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromDirectProductToExponential. F: ( a, b ) \mapsto \mathtt{MorphismFromDirectProductToExponential}(a, b).

 ‣ AddMorphismFromDirectProductToExponentialWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismFromDirectProductToExponentialWithGivenObjects. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromDirectProductToExponentialWithGivenObjects}(s, a, b, r).

 ‣ AddMorphismToCartesianBidual( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismToCartesianBidual. F: ( a ) \mapsto \mathtt{MorphismToCartesianBidual}(a).

 ‣ AddMorphismToCartesianBidualWithGivenCartesianBidual( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MorphismToCartesianBidualWithGivenCartesianBidual. F: ( a, r ) \mapsto \mathtt{MorphismToCartesianBidualWithGivenCartesianBidual}(a, r).

 ‣ AddUniversalPropertyOfCartesianDual( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalPropertyOfCartesianDual. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfCartesianDual}(t, a, alpha).

 ‣ AddLeftCartesianDistributivityExpanding( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftCartesianDistributivityExpanding. F: ( a, L ) \mapsto \mathtt{LeftCartesianDistributivityExpanding}(a, L).

 ‣ AddLeftCartesianDistributivityExpandingWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftCartesianDistributivityExpandingWithGivenObjects. F: ( s, a, L, r ) \mapsto \mathtt{LeftCartesianDistributivityExpandingWithGivenObjects}(s, a, L, r).

 ‣ AddLeftCartesianDistributivityFactoring( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftCartesianDistributivityFactoring. F: ( a, L ) \mapsto \mathtt{LeftCartesianDistributivityFactoring}(a, L).

 ‣ AddLeftCartesianDistributivityFactoringWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation LeftCartesianDistributivityFactoringWithGivenObjects. F: ( s, a, L, r ) \mapsto \mathtt{LeftCartesianDistributivityFactoringWithGivenObjects}(s, a, L, r).

 ‣ AddRightCartesianDistributivityExpanding( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightCartesianDistributivityExpanding. F: ( L, a ) \mapsto \mathtt{RightCartesianDistributivityExpanding}(L, a).

 ‣ AddRightCartesianDistributivityExpandingWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightCartesianDistributivityExpandingWithGivenObjects. F: ( s, L, a, r ) \mapsto \mathtt{RightCartesianDistributivityExpandingWithGivenObjects}(s, L, a, r).

 ‣ AddRightCartesianDistributivityFactoring( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightCartesianDistributivityFactoring. F: ( L, a ) \mapsto \mathtt{RightCartesianDistributivityFactoring}(L, a).

 ‣ AddRightCartesianDistributivityFactoringWithGivenObjects( C, F ) ( operation )
The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation RightCartesianDistributivityFactoringWithGivenObjects. F: ( s, L, a, r ) \mapsto \mathtt{RightCartesianDistributivityFactoringWithGivenObjects}(s, L, a, r).