\begin{definition}[Definition:Contravariant Hom Functor]
Let $\mathbf{Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $C \in \mathbf C_0$ be an object of $\mathbf C$.
The '''contravariant hom functor based at $C$''':
: $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$
is the covariant functor defined by:
{{begin-axiom}}
{{axiom | lc= Object functor:
        | m = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right)
}}
{{axiom | lc= Morphism functor:
        | m = \operatorname{Hom}_{\mathbf C} \left({f, C}\right): \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \to \operatorname{Hom}_{\mathbf C} \left({A, C}\right), g \mapsto g \circ f
        | rc= for $f: A \to B$
}}
{{end-axiom}}
where $\operatorname{Hom}_{\mathbf C} \left({B, C}\right)$ denotes a hom set.
Thus, the morphism functor is defined to be precomposition.
\end{definition}