Hom-Functor

Let be Locally Small.
We define the Covariant hom-functor to be:

We send each object to the Hom-Set
We send each morphism to a function

defined by

Similarly, the Contravariant hom-functor

sends to
and to the map
given by

Lemma

The Covariant hom-functor is a Functor.
Similarly, the Contravariant hom-functor is a Contravariant Functor.

Proof

Functoriality follows from the Associativity law in

Lemma

Let be a Morphism in .
Then it induces a Natural Transformation

given by for any
Dually,

is a Natural Transformation given by for .

Proof

Let (in )
Consider the diagram

Let .
Clearly:

by Associativity in .
Thus

for any .
Thus is a Natural Transformation.