# Category of functors

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

## The category of functors

Let ${\displaystyle C}$ and ${\displaystyle D}$ be two categories. The category of functors ${\displaystyle Funct(C^{op},Sets)}$ has

1. Objects are functors ${\displaystyle F:C^{op}\to D}$
2. A morphism of functors ${\displaystyle F,G}$ is a natural transformation ${\displaystyle \eta :F\to G}$; i.e., for each object ${\displaystyle U}$ of ${\displaystyle C}$, a morphism in ${\displaystyle D}$ ${\displaystyle \eta _{U}:F(U)\to G(U)}$ such that for all morphisms ${\displaystyle f:U\to V}$ in ${\displaystyle C^{op}}$, the diagram (DIAGRAM) commutes.

A natural isomorphism is a natural transformation ${\displaystyle \eta }$ such that ${\displaystyle \eta _{U}}$ is an isomorphism in ${\displaystyle D}$ for every object ${\displaystyle U}$. 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 ${\displaystyle h_{X}=Mor_{C}(-,X)}$.

## Examples

1. In the theory of schemes, the presheaves ${\displaystyle h_{X}}$ 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 ${\displaystyle C}$ be a category and let ${\displaystyle X,X'}$ be objects of ${\displaystyle C}$. Then

1. If ${\displaystyle F}$ is any contravariant functor ${\displaystyle F:C^{op}\to Sets}$, then the natural transformations of ${\displaystyle Mor_{C}(-,X)}$ to ${\displaystyle F}$ are in correspondence with the elements of the set ${\displaystyle F(X)}$.
2. If the functors ${\displaystyle Mor_{C}(-,X)}$ and ${\displaystyle Mor_{C}(-,X')}$ are isomorphic, then ${\displaystyle X}$ and ${\displaystyle X'}$ are isomorphic in ${\displaystyle C}$. More generally, the functor ${\displaystyle h:C\to Funct(C^{op},Sets)}$, ${\displaystyle X\mapsto h_{X}}$, is an equivalence of categories between ${\displaystyle C}$ and the full subcategory of representable functors in ${\displaystyle Funct(C^{op},Sets)}$.