Definition:Yoneda Functor/Contravariant

Definition
Let $C$ be a locally small category.

Let $\mathbf{Set}$ be the category of sets.

Let $\left[{C, \mathbf{Set} }\right]$ be the functor category between them.

The contravariant Yoneda functor of $C$ is the contravariant functor $h^- : C \to \left[{C, \mathbf{Set} }\right]$ which sends
 * an object $X \in C$ to the covariant hom-functor $h^X = \operatorname{Hom} \left({X, -}\right)$
 * a morphism $f : X \to Y$ to the precomposition natural transformation $h^f : \operatorname{Hom} \left({Y, -}\right) \to \operatorname{Hom} \left({X, -}\right) : g \mapsto g \circ f$

Also see

 * Yoneda Lemma
 * Definition:Covariant Yoneda Functor
 * Definition:Hom Functor