contravariant representation of a functor

Let \(\mathcal{C}\) be a locally small category and \(\mathrm{Set}\) the category set and \(\mathcal{F}: \mathcal{C}^{\mathrm{op}} \rightarrow \mathrm{Set}\) be a contravariant functor

Then a representation of \(\mathcal{F}\) is an object \(A \in \mathrm{Ob}(\mathcal{C})\) and a natural isomorphism

\begin{align*} \eta: \mathcal{F} \rightarrow \mathrm{Hom}_{\mathcal{C}}(-,A) \end{align*}

see:

• contravariant Yoneda