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

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

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 Embedding Theorem
 * Definition:Contravariant Yoneda Functor