# Definition:Yoneda Functor/Yoneda Embedding

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## 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.

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 $h_f : \operatorname{Hom} \left({-, X}\right) \to \operatorname{Hom} \left({-, Y}\right)$

## Also denoted as

The **Yoneda embedding** is also denoted by $Y$.

## Also known as

The **Yoneda embedding** can also be referred to as the **(covariant) Yoneda functor**.

## Also see

## Source of Name

This entry was named for Nobuo Yoneda.