# Category of functors

From Citizendium

This article focuses on the category of contravariant functors between two categories.

## The category of functors

Let and be two categories. The category of functors has

- Objects are
**functors** - A morphism of functors is a
**natural transformation**; i.e., for each object of , a morphism in such that for all morphisms in , the diagram (DIAGRAM) commutes.

A *natural isomorphism* is a natural transformation such that is an isomorphism in for every object . One can verify that natural isomorphisms are indeed isomorphisms in the category of functors.

An important class of functors are the *representable* functors; i.e., functors that are naturally isomorphic to a functor of the form .

## Examples

- In the theory of schemes, the presheaves are often referred to as the
*functor of points*of the scheme*X*. Yoneda's lemma allows one to think of a scheme as a functor in some sense, which becomes a powerful interpretation; indeed, meaningful geometric concepts manifest themselves naturally in this language, including (for example) functorial characterizations of smooth morphisms of schemes.

## The Yoneda lemma

Let be a category and let be objects of . Then

- If is any contravariant functor , then the natural transformations of to are in correspondence with the elements of the set .
- If the functors and are isomorphic, then and are isomorphic in . More generally, the functor , , is an equivalence of categories between and the full subcategory of
*representable*functors in .