nLab
corepresentable functor

A functor $C\to \mathrm{Set}$ is corepresentable if it is (isomorphic to) a functor of the form ${\mathrm{Hom}}_C(c,-)$ for some object $c\in C$.

This is equivalently a representable functor defined on the opposite category $C^{\mathrm{op}}$. Often no terminological distinction is made between representable and corepresentable ones (both being called simply “representable”), since a functor $C\to \mathrm{Set}$ can only be “corepresentable” while a functor $C^{\mathrm{op}}\to \mathrm{Set}$ can only be “representable”.