Yoneda Lemma

Yoneda Embedding

The Yoneda embedding is the functor taking to the contravariant representable functor: and taking to the natural transformation:

Yoneda Lemma

Let be a locally small category. For any object and functor , there is an isomorphism which is natural in both and .

• Naturality in means that, given any , the following diagram commutes:
• Naturality in means that, given any , the following diagram commutes:

Theorem

The Yoneda embedding is full and faithful.

Proof For any objects , we have an isomorphism: And this isomorphism is indeed induced by the functor , since it takes an element of to the natural transformation given by where has component at :So, .

Corollary Yoneda Principle Given objects and in any locally small category , implies , where is the Yoneda embedding.

Prop If the cartesian closed category has coproducts, then is distributive, that is, there is always a canonical isomorphism: