gap> G := SymmetricGroup( 3 );;
gap> CG := GroupAsCategory( G );;
gap> u := GroupAsCategoryUniqueObject( CG );;
gap> SetOfObjectsOfCategory( CG ) = [ u ];
true
gap> Length( SetOfMorphismsOfFiniteCategory( CG ) ) = Size( G );
true
gap> x := (2,3)/CG;;
gap> id := ()/CG;;
gap> IsIdenticalObj( x * x, id );
true
gap> alpha := GroupAsCategoryMorphism( (1,2,3), CG );;
gap> alpha * Inverse( alpha ) = IdentityMorphism( u );
true
gap> beta := GroupAsCategoryMorphism( (1,2,3,5), CG );;
gap> IsWellDefined( beta );
false
gap> gamma := GroupAsCategoryMorphism( (1,3), CG );;
gap> IsWellDefined( gamma );
true
gap> Lift( alpha, gamma ) * gamma = alpha;
true
gap> alpha * Colift( alpha, gamma ) = gamma;
true
#@if IsPackageMarkedForLoading( "FinSetsForCAP", ">= 2023.07-03" )
gap> Length( HomomorphismStructureOnObjects( u, u ) ) = Size( G );
true
gap> InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism(
>     u,u,
>     PreCompose(
>         InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure( alpha ), HomomorphismStructureOnMorphisms( gamma, Inverse( gamma ) )
>     )
> )
> =
> gamma * alpha * Inverse( gamma );
true
#@fi