CatGloss

An easy application of the Yoneda lemma completely characterizes natural transformations between representable functors: any locally small category C\mathsf{C} is isomorphic to the full subcategory of catCop\textup{\textsf{cat}}^{\mathsf{C}^\mathrm{op}} spanned by the contravariant represented functors, and Cop\mathsf{C}^\mathrm{op} is isomorphic to the full subcategory of catC\textup{\textsf{cat}}^{\mathsf{C}} spanned by the covariant represented functors, via the canonically-defined covariant and contravariant Yoneda embeddings.