Representation

Suppose a Functor is Representable.
Then is Isomorphic to some .
We say that is a representation of
where is a Natural Isomorphism

By Yoneda Lemma, there is some element such that

for any .
In this case, we also say that is a representation of .
We also call a Universal Element.

Lemma

If and are both representations of
then there is a unique Isomorphism in such that .

Proof

Let in
Let be a Natural Isomorphism such that
and let be a Natural Isomorphism such that .
Now

Thus
if and only if

Clearly now is the unique

There is also this Commutative Diagram