Contravariant hom functor

Let $\mathcal{C}$ be a locally small category. For each object $B$ in $\mathcal{C}$, there is a functor
$$\operatorname{Hom}_{\mathcal{C}}(-,B):\mathcal{C}^{\operatorname{op}}\rightarrow \mathbf{Set}$$defined by:

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