CatGloss

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