For a locally small category, its hom-functor is the functor
from the product category of the category with its opposite category to the category Set of sets, which sends
an object , i.e. a pair of objects in , to the hom-set in , the set of morphisms in ;
a morphism , i.e. a pair of morphisms
in to the function that sends
Note: when the symbol is used, it denotes traditional right-to-left order of composition. For those who prefer the left-to-right order, the symbol may be used in place of . Further discussion of this should go to the nForum page here.
More generally, for a closed symmetric monoidal category and a -enriched category, its hom-functor is the functor
that sends objects to the hom-object .
Some categories are equipped with an operation that behaves like a hom-functor, but takes values in itself
Such an operation is called an internal hom functor, and categories carrying this are called closed categories.
Given a hom-functor , for any object one obtains a functor
given by and a functor
given by , i.e. by fixing one of the arguments of to be .
Formally this is
Functors of the form are called presheaves on , and functors naturally isomorphic to are called representable functors or representable presheaves on .
Functors of the form are called copresheaves on , and functors naturally isomorphic to are called corepresentable functors or representable copresheaves on .
Preservation of limits
The hom-functor preserves all limits in both arguments separately. This means:
for fixed object the functor sends limit diagrams in to limit diagrams in ;
for fixed object the functor sends limit diagrams in – which are colimit diagrams in ! – to limit diagrams in .
For instance for
a pullback diagram in and for any object, the induced diagram
in Set is again a pullback diagram. A moment of reflection shows that this statement is equivalent to the very definition of limit.
Relation to profunctors
The hom-functor is also the identity profunctor .
One way to see this is to notice that its adjunct
under the internal hom adjunction in the 1-category Cat is the functor
where is the Yoneda embedding. Profunctors whose hom-adjunct is of the form for an ordinary functor are those in the inclusion of these ordinary functors into profunctors. So the hom-functor is the image of the identity functor under this inclusion.