Definition:Representation of Functor

Definition
Let $\mathbf C$ be a locally small category.

Let $\mathbf{Set}$ be the category of sets.

Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor.

A representation of $F$ is a pair $\tuple {C, \eta}$ where $\eta : \operatorname{Hom} \paren{ C, \cdot } \to F$ is a natural isomorphism with the covariant hom functor of $C$.

Also see

 * Definition:Representable Functor
 * Uniqueness of Representing Objects