# Yoneda Lemma

The **Yoneda lemma** is a result in category theory.

## Contents

## Statement

Let be a locally small category, and let be
a functor from to **Set**, the category of sets.
Let denote the functor that
sends every object to and
that takes the arrow to the function
given by
. In other words, is the
hom functor . Then there exists a bijection
between the set of natural transformations from to
and the set . In symbols,

## Proof

Let be a natural transformation.
Then for each object of , gives us an arrow
in the category **Set**, i.e., a
function , such that the diagram
commutes for any arrow in .
In particular, we have
But is the map from
to . We thus have
Thus for every object of , the morphism
is uniquely determined by the element .
Thus the map is an injection
from to .

It thus remains to be shown that for any , the maps for every object of define a natural transformation. But this is true, as for any objects and of , any morphism of , and any element of , we have since is a functor. Thus the diagram commutes, so is a natural transformation, as desired.

Therefore the map is a bijection between and , as desired.

## Dual Statement

If we replace the category with its opposite category
, then we get the following
result: if is a contravariant functor from
to **Set**, then
for every object of . This dual statement
is also sometimes known as the Yoneda lemma.