\begin{definition}[Definition:Covariant 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 '''covariant hom functor based at $C$''', $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right): \mathbf C \to \mathbf{Set}$, is the covariant functor defined by:
{{begin-axiom}}
{{axiom|lc= Object functor:
       |m = \operatorname{Hom}_{\mathbf C} \left({C, D}\right) = \operatorname{Hom}_{\mathbf C} \left({C, D}\right)
}}
{{axiom|lc= Morphism functor:
       |m = \operatorname{Hom}_{\mathbf C} \left({C, f}\right): \operatorname{Hom}_{\mathbf C} \left({C, A}\right) \to \operatorname{Hom}_{\mathbf C} \left({C, B}\right), g \mapsto f \circ g
       |rc= for $f: A \to B$
}}
{{end-axiom}}
where $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$ denotes a hom set.
Thus, the morphism functor is defined to be postcomposition.
That $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right)$ is a functor is shown on Covariant Hom Functor is Functor.
\end{definition}