1 Yoneda Lemma

We first recall a standard criterion for equivalence of categories.

Lemma (Equivalence criterion). Let 𝒜 and ℬ be categories, and let

be a fully faithful and essentially surjective functor. Then F is an equivalence of categories. In particular, this applies when 𝒜 and ℬ are Abelian categories and F is exact.

We next introduce the Hom-functors. Let Set denote the category of sets. For a category 𝒞 and an object X ∈𝒞, fixing the first argument of the Hom bifunctor gives a covariant functor

The assignment

is contravariant in X. Equivalently, it defines a covariant functor

where [𝒞,Set] denotes the functor category.

Dually, the usual Yoneda embedding is

Theorem (Yoneda lemma). Let F : 𝒞^op → Set be a functor and let X ∈𝒞. Then there is a natural bijection

The bijection sends a natural transformation η to η_X(id _X).

As a standard consequence of the Yoneda lemma, the Yoneda embedding Y is fully faithful. More explicitly, for any objects X,Y ∈𝒞 there is a natural bijection

Thus morphisms in 𝒞 are recovered exactly as natural transformations between the corresponding representable functors.