Representable contravariant functor

Definition

Suppose $\mathcal{C}$ is a category and $\mathcal{F}:\mathcal{C}\to \mathrm{S}\mathrm{e}\mathrm{t}$ is a contravariant functor to the category of sets. We say that $\mathcal{F}$ is representable if there exists a natural isomorphism between $\mathcal{F}$ and the contravariant Hom functor corresponding to some object $A\in \mathrm{O}\mathrm{b}\mathcal{C}$.

There is a related notion of representable functor, used for (covariant) functors.