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

The Yoneda embedding of $C$ is the covariant functor $h_- : C \to [C^{\operatorname{op}}, \mathbf{Set}]$ which sends
 * an object $X\in C$ to the contravariant hom-functor $h_X = \operatorname{Hom}(-, X)$
 * a morphism $f : X \to Y$ to the induced natural transformation $h_f : \operatorname{Hom}(-, X) \to \operatorname{Hom}(-, Y)$

Also denoted as
The Yoneda embedding is also denoted by $Y$.

Also see

 * Yoneda Lemma