Yoneda embedding

Summary

The Yoneda embedding of a locally small category $\mathcal{C}$ is a functor into presheaves. It sends each object $A$ to its hom functor $\Hom(-, A)$, a presheaf on $\mathcal{C}$. By the Yoneda lemma it is fully faithful, so $\mathcal{C}$ is equivalent to a full subcategory of a category of presheaves. This lets one study objects of any category through set-valued functors, where many constructions are easy.