Definition:Representable 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.

Then $F$ is representable there exists an object $C \in \mathbf C$ such that $F$ is naturally isomorphic to the covariant hom functor $\map {\operatorname{Hom} } {C, \cdot}$.

That is, $F$ is representable $F$ has a representation.

Also see

 * Uniqueness of Representing Objects
 * Definition:Hom Functor
 * Definition:Representation of Functor
 * Definition:Contravariant