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
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C}(B,\cdot) \arrow[rr,"\mathcal{C}(f{,}\cdot)"] \arrow[dr,swap,"\beta"]
&&
\mathcal{C}(A,\cdot) \arrow[dl,"\alpha"] \\
& F
\end{tikzcd}
\end{document}