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 $(A, \eta)$ where $\eta : \operatorname{Hom}(A, -) \to F$ is a natural isomorphism with the covariant hom functor of $A$.

Also see

 * Definition:Representable Functor
 * Uniqueness of Representing Objects