Definition:Representable Functor

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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 if there exists an object $a\in \mathbf C$ such that $F$ is naturally isomorphic to the covariant hom functor $\operatorname{Hom}(a,-)$.

Also see