Representable contravariant functor

Definition

Suppose ${\mathcal {C}}$ is a category and ${\mathcal {F}}:{\mathcal {C}}\to \operatorname {Set}$ 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 $A\in \operatorname {Ob} {\mathcal {C}}$.

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