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

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C}(B,C) \arrow[r,"\mathcal{C}(B{,}g)"] \arrow[d,swap,"\mathcal{C}(f{,}C)"]
 & \mathcal{C}(B,D) \arrow[d,"\mathcal{C}(f{,}D)"] \\
\mathcal{C}(A,C) \arrow[r,"\mathcal{C}(A{,}g)"]
 & \mathcal{C}(A,D)
\end{tikzcd}
\end{document}

Let . Clearly:

by Associativity in . Thus

for any . Thus is a Natural Transformation.