$\quad$\n1. A covariant or contravariant functor $F$ from a locally small category $\mathsf{C}$ to $\textup{\textsf{cat}}$ is representable if there is an object $c \in \mathsf{C}$ and a natural isomorphism between $F$ and the functor of appropriate variance represented by $c$, in which case one says that the functor $F$ is represented by the object $c$.\n2. A representation for a functor $F$ is a choice of object $c \in \mathsf{C}$ together with a specified natural isomorphism $\mathsf{C}(c,-) \cong F$, if $F$ is covariant, or $\mathsf{C}(-,c)\cong F$, if $F$ is contravariant.