Definition:Yoneda Functor/Yoneda Embedding

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

 * Yoneda Lemma
 * Definition:Contravariant Yoneda Functor