Yoneda Lemma for Covariant Functors

Theorem

Let $C$ be a locally small category.

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

Bijection

Let $F: C \to \mathbf {Set}$ be a covariant functor.

Let $A \in C$ be an object.

Let $I_A$ be its identity morphism.

Let $h^A = \map {\operatorname {Hom} } {A, -}$ be its covariant hom-functor.

The class of natural transformations $\map {\operatorname {Nat} } {h^A, F}$ is a small class, and:

$\alpha: \map {\operatorname {Nat} } {h^A, F} \to \map F A: \eta \mapsto \map {\eta_A} {I_A}$

$\beta: \map F A \to \map {\operatorname {Nat} } {h^A, F}: u \mapsto \paren {X \mapsto \paren {f \mapsto \map {\paren {\map F f} } u} }$

are inverses of each other.

Naturality

Let $\sqbrk {C, \mathbf {Set} }$ be the covariant functor category.

Let $C \times \sqbrk {C, \mathbf {Set} }$ be the product category.

Let $C \times \sqbrk {C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A, F} \mapsto \map {\operatorname {Nat} } {h^A, F}$ be the covariant functor defined as the composition of the hom bifunctor and the product of the contravariant Yoneda functor $h^-$ and the identity functor $\operatorname{id}_{\sqbrk {C, \mathbf {Set} } }$.

Let $\operatorname{ev} : C \times \sqbrk {C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A, F} \mapsto \map F A$ be the functor evaluation functor.

Then $\Phi_{\tuple {A, F} } : \map {\operatorname {Nat} } {h^A, F} \to \map F A: \eta \mapsto \map {\eta_A} {\operatorname {id}_A}$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.

Also see

• Yoneda Lemma for Contravariant Functors

Source of Name

This entry was named for Nobuo Yoneda.