gap> q := RightQuiver( "q(1)[a:1->1]" );
q(1)[a:1->1]
gap> C2 := Category( q, [ [ q.a^2, q.1 ] ] );
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a = 1 ]
gap> PSh := PreSheaves( C2 );
PreSheaves( FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )/ [ a*a = 1 ],
SkeletalFinSets )
gap> IsReflexive( TerminalObject( PSh ) );
true
gap> IsReflexive( InitialObject( PSh ) );
true
gap> L := PSh.1;
<A projective object in PreSheaves( FreeCategory(
 RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a = 1 ], SkeletalFinSets )>
gap> IsReflexive( L );
true
gap> P := DirectProduct( L, L );
<An object in PreSheaves( FreeCategory(
 RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a = 1 ], SkeletalFinSets )>
gap> IsReflexive( P );
true
gap> iota := UnitOfIsbellAdjunction( PSh );
A natural transformation from Identity functor of PreSheaves(
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a = 1 ], SkeletalFinSets )
to Precomposition of Isbell left adjoint functor and Isbell right adjoint functor
gap> iotaP := iota( P );
<A morphism in PreSheaves( FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
 / [ a*a = 1 ], SkeletalFinSets )>
gap> Display( iotaP );
Image of <(1)>:
{ 0,..., 3 } ⱶ[ 0, 1, 2, 3 ]→ { 0,..., 3 }

A morphism in PreSheaves( FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
/ [ a*a = 1 ], SkeletalFinSets ) given by the above data
gap> IsOne( iotaP );
true
gap> C := Coproduct( L, L );
<An object in PreSheaves( FreeCategory(
 RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a = 1 ], SkeletalFinSets )>
gap> IsReflexive( C );
true
gap> iotaC := iota( C );
<A morphism in PreSheaves( FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
 / [ a*a = 1 ], SkeletalFinSets )>
gap> Display( iotaC );
Image of <(1)>:
{ 0,..., 3 } ⱶ[ 1, 2, 0, 3 ]→ { 0,..., 3 }

A morphism in PreSheaves( FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
/ [ a*a = 1 ], SkeletalFinSets ) given by the above data
gap> IsIsomorphism( iotaC );
true
gap> Source( iotaC ) = C;
true
gap> Range( iotaC ) = P;
true
gap> IsReflexive( Coproduct( L, L, L ) );
false

gap> q := RightQuiver( "q(1)[t:1->1]" );
q(1)[t:1->1]
gap> F := FreeCategory( q );
FreeCategory( RightQuiver( "q(1)[t:1->1]" ) )
gap> Q := HomalgFieldOfRationals( );
Q
gap> Qq := Q[F];
Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) )
gap> A := Qq / [ Qq.t^3 ];
Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) / relations
gap> PSh := PreSheaves( A );
PreSheaves(
Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) / relations,
Rows( Q ) )
gap> CommutativeRingOfLinearCategory( PSh );
Q
gap> Display( PSh.1 );
Image of <(1)>:
A row module over Q of rank 3

Image of (1)-[{ 1*(t) }]->(1):
Source:
A row module over Q of rank 3

Matrix:
[ [  0,  1,  0 ],
  [  0,  0,  1 ],
  [  0,  0,  0 ] ]

Range:
A row module over Q of rank 3

A morphism in Rows( Q )

An object in PreSheaves(
Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) / relations,
Rows( Q ) ) given by the above data
gap> Display( PSh.t );
Image of <(1)>:
Source:
A row module over Q of rank 3

Matrix:
[ [  0,  1,  0 ],
  [  0,  0,  1 ],
  [  0,  0,  0 ] ]

Range:
A row module over Q of rank 3

A morphism in Rows( Q )

A morphism in PreSheaves(
Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) / relations,
Rows( Q ) ) given by the above data
gap> phi := PreCompose( PSh.t, PSh.t );
<(1)->3x3>
gap> V := DirectSum( [ PSh.1, KernelObject( phi ), ImageObject( phi ) ] );
<(1)->6; (t)->6x6>
gap> Display( V );
Image of <(1)>:
A row module over Q of rank 6

Image of (1)-[{ 1*(t) }]->(1):
Source:
A row module over Q of rank 6

Matrix:
[ [  0,  1,  0,  0,  0,  0 ],
  [  0,  0,  1,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  1,  0 ],
  [  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0 ] ]

Range:
A row module over Q of rank 6

A morphism in Rows( Q )

An object in PreSheaves(
Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) / relations,
Rows( Q ) ) given by the above data
gap> IsProjective( V );
false
gap> IsReflexive( V );
true