Let be a small category. For a presheaf , let denote the category of elements of , whose elements are pairs and whose morphisms are maps such that . There is a natrual projection , and we denote the Yoneda embedding as .
Theorem: For any presheaf , we have:
Proof: We will construct a limiting cocone over the diagram with vertex . The cone will be
where is defined to be the natural transformation that corresponds to under the Yoneda correspondence. Precisely, . To confirm that this actually forms a cone, we must show that for a map that . Indeed, for any , we have:
Where the values are equal since is a contravariant functor. Now let
be another cocone. We must find a universal map so that . Define by letting be precisely the element of that corresponds to under the Yoneda correspondence. Explicitly, . We confirm that this is actually a natural transformation, by considering the naturality square for a map :
Now, since can also be seen as a map in , we have
since the ’s form a cocone. On the other hand, since is a natural transformation itself, the diagram
commutes, and we have
and the two expressions are equal.
Next, we must check that . Consider:
The two expressions are equal due to the fact that is a map in and the ’s form a cone:
Finally, for uniqueness, suppose that is another map with . Recall that for any , we have (the Yoneda correspondence). As such, for any :
where the last step is by definition. Thus, .
Therefore, is the vertex of a limiting cocone over , showing the claim.