representing object

Let \(\mathcal{C}\) be a locally small category, \(\mathcal{F}: \mathcal{C} \rightarrow \mathrm{Set}\) a covariant resp. contravariant functor. Then a representing object is an \(A \in \mathrm{Ob}(\mathcal{C})\) such that

\begin{align*} \mathcal{F} \cong& \mathrm{Hom}_{\mathcal{C}}(A,-) \\ \mathcal{F} \cong \mathrm{Hom}_{\mathcal{C}}(-,A) \end{align*}

depending whether \(\mathcal{F}\) is co/contravariant