\begin{definition}[Definition:Representation of Functor]
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 $C$.
\end{definition}