Representable functor

Definition

Suppose $\mathcal{C}$ is a category, and $\mathcal{F}:\mathcal{C}\to \mathrm{S}\mathrm{e}\mathrm{t}$ 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 $A\in \mathrm{O}\mathrm{b}\mathcal{C}$ 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.