Hom functor

Summary

In a locally small category $\mathcal{C}$, each object $A$ gives a functor $\Hom(A, -)$ from $\mathcal{C}$ to the category of sets. It sends an object $X$ to the set of morphisms $A \to X$. Dually, $\Hom(-, A)$ is a contravariant functor. Hom functors are the basic examples of functors into sets, and the Yoneda lemma says that they control all other such functors.