Representable functor

Definition

Suppose ${\mathcal {C}}$ is a category, and ${\mathcal {F}}:{\mathcal {C}}\to \operatorname {Set}$ is a functor. In other words, ${\mathcal {F}}$ is a functor from ${\mathcal {C}}$ to the category of sets. We say that ${\mathcal {F}}$ is representable if there exists an object $A\in \operatorname {Ob} {\mathcal {C}}$ such that there is a natural isomorphism between ${\mathcal {F}}$ and the covariant Hom functor corresponding to the object $A$.

There is an analogous notion of representable contravariant functor.