covariant hom functor of points

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

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

see: hom-functor