# Definition:Yoneda Functor

## Definition

Let $C$ be a locally small category.

Let $C^{\operatorname{op}}$ be its opposite category.

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

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

### Yoneda Embedding

The Yoneda embedding of $C$ is the covariant functor $h_- : C \to \left[{C^{\operatorname{op}}, \mathbf{Set} }\right]$ which sends:

an object $X\in C$ to the contravariant hom-functor $h_X = \operatorname{Hom} \left({-, X}\right)$
a morphism $f : X \to Y$ to the postcomposition natural transformation $h_f : \operatorname{Hom} \left({-, X}\right) \to \operatorname{Hom} \left({-, Y}\right)$

### Contravariant Yoneda Functor

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 $h^f : \operatorname{Hom} \left({Y, -}\right) \to \operatorname{Hom} \left({X, -}\right) : g \mapsto g \circ f$

## Source of Name

This entry was named for Nobuo Yoneda.