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For questions about and related to Representable Functors, that are set-valued functors which can be "represented" by the hom-set of a single object from the domain category. Should be used together with the (category-theory) tag.

A covariant functor $F:{\rm C}\to{\rm Set}$ is said to be representable, if there is an object $c$ of the domain category ${\rm C}$ such that $F$ is naturally isomorphic to ${\rm Hom}(c,-)$. A representation of $F$ is a choice of an object $c$ together with a specified natural isomorphism.

A contravariant functor $F:{\rm C^{op}}\to{\rm Set}$ (i.e. a presheaf) is said to be representable, if there is an object $c$ of the domain category ${\rm C}$ such that $F$ is naturally isomorphic to ${\rm Hom}(-,c)$. A representation of $F$ is a choice of an object $c$ together with a specified natural isomorphism.