I'm studying this text which provides a shortcut in category theory for the notions I'm interestd in. Consider this theorem:
I understand the left to right implication. However, I'm not sure what is implied by the right to left implication.
To me it seems like they're taking any bijection between $hom(D_0,-)$ and $hom(C,G -)$ and proving it has to be of the form $G - \circ \; u$.
On the other hand there seems to be an abuse of notation as $u = \varphi(id_{D_0})$ seems to suggest that we can apply $\varphi$ to morphisms of $D$ while I would be expecting that only a particular $\varphi_D$ component of the natural isomorphism could be applied to it.
Could you please clarify what is implied by the theorem? A slow proof might enlighten this.
It's easier to use a commutative diagram, to see what is going on. Let's paste the one from Wiki and use it to prove the claim.
The definition implies that the function $\Phi:\hom[A,Y]\to \hom[X,U(Y)]:g\mapsto U(g)\circ \phi$ is a natural isomorphism, as you point out.
On the other hand, suppose there is a natural isomorphism $\Phi:\hom[A,Y]\to \hom[X,U(Y)]$. You want to prove that $\Phi(1_{A}):X\to U(A)$ is universal. So, let $Y\in D$ and $f: X \to U(Y)$, and define $g=\Phi^{-1}(f).$ Then, we get what we want because, using the naturality of $\Phi$ (where?), we have
$$U(g)\circ \Phi(1_A)=U(\Phi^{-1}(f))\circ \Phi(1_A)=\Phi(g)=\Phi\circ \Phi^{-1}(f)=f$$
Uniqueness of $g$ follows immediately because $\Phi$ is an isomorphism.