contravariant hom-functor of points

Let \(\mathcal{C}\) be a locally small category and \(A \in \mathcal{C}\) Then the contravariant hom-functor is defined as

\begin{align*} \mathrm{Hom}(-, A): \mathcal{C} \rightarrow& \mathrm{Set} \\ B \mapsto& \mathrm{Hom}_{\mathrm{Set}}(B, A) \\ (f: B \rightarrow C) \mapsto& (\mathrm{Hom}(f,B): g \mapsto g \circ h) \end{align*}

see: hom-functor