Covariant hom functor

Let $\mathcal{C}$ be a locally small category. For each object $A$ in $\mathcal{C}$, there is a functor

$$\operatorname{Hom}_{\mathcal{C}}(A,-):\mathcal{C}\rightarrow \mathbf{Set}$$

defined by:

• On objects: each object $X$ is sent to the set $\operatorname{Hom}_{\mathcal{C}}(A,X)$.
• On morphisms: a morphism $f:X\rightarrow Y$ is sent to the function
  $$f_*=\operatorname{Hom}_{\mathcal{C}}(A, f): \operatorname{Hom}_{\mathcal{C}}(A, X) \rightarrow \operatorname{Hom}_{\mathcal{C}}(A, Y)$$given by post-composition: $g\mapsto f\circ g$.