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,
![\[\text{Nat}(h_A, F) \cong F(A) .\]](http://latex.artofproblemsolving.com/c/3/f/c3fb159ef6dc888cd40970345eadd8b84a883f43.png)
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
![\[\begin{array}{ccc} h_A(A) & \!\stackrel{\phi_A}{\longrightarrow}\! & F(A) \\ \downarrow h_A f && \downarrow Ff \\ h_A(B) & \! \stackrel{\phi_B}{\longrightarrow}\! & F(B) \end{array}\]](http://latex.artofproblemsolving.com/b/7/d/b7dc3270aa9ef95caf76f5d4889784af3e91c0af.png)
commutes for any arrow 
in .
In particular, we have
![\[(\phi_B h_A f)(\text{Id}_A) = (Ff \phi_A) ( \text{Id}_A) .\]](http://latex.artofproblemsolving.com/2/e/8/2e81fcd66f761595d611bf2384a86b2bf8ca2d97.png)
But 
is the map from
to . We thus have
![\[\phi_B (f) = \phi_B (f \circ \text{Id}_A) = Ff (\phi_A \text{Id}_A) .\]](http://latex.artofproblemsolving.com/d/3/b/d3b9dc60bcb5cfde4cebbd64d074e3e6bcd23742.png)
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
![\[\phi_B : f \mapsto F f x\]](http://latex.artofproblemsolving.com/5/8/d/58dfb00dca57bfcafe3c5a8b212bcf94e833f407.png)
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
![\[\phi_C h_A f (\alpha) = \phi_C f \alpha = F (f \alpha) x = (F f )(F\alpha) x = (F f) \phi_B (\alpha) ,\]](http://latex.artofproblemsolving.com/6/5/b/65bd51d7f9f074c821a6adb9b3266948b85767f7.png)
since is a functor. Thus the diagram
![\[\begin{array}{ccc} h_A(B) & \!\stackrel{\phi_B}{\longrightarrow}\! & F(B) \\ \downarrow h_A f && \downarrow F f \\ h_A (C) & \!\stackrel{\phi_C}{\longrightarrow}\! & F(C) \end{array}\]](http://latex.artofproblemsolving.com/0/5/e/05e26a4e5821bf87e855c5d1b06cb94889c09682.png)
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
![\[\text{Nat} \bigl( \text{Hom}( - ,A), F \bigr) \cong F(A)\]](http://latex.artofproblemsolving.com/e/e/e/eee36e0ec24b969125c8a287ca0f7e0810329cfe.png)
for every object 
of . This dual statement
is also sometimes known as the Yoneda lemma.
