I'm using MacLane as a reference for some of the category theory sections in notes I'm writing on simplicial sets and came across this proof but have been unable to understand it (It is on page 76 of the second edition). I think (in my opinion) bad notation and a few too many blank spaces along with inexperience with Yoneda's lemma has left me unable to decode this proof. I've done my best at cleaning up the notation in my own write up, but there are still two things I don't understand. First, here's what I've written so far:
As noted in the proof, I cannot figure out why $\eta$ is well defined. Since $x$ is the only important bit of information you should get the same $z$ regardless of what function we chose. I have no idea how to prove this, though.
My other problem is that MacLane finishes his proof at the point I have, and says that $\eta$ being natural follows from $G(f)(Y_G(z))=Y_G(G(f)(z))=Y_G(z')$. Yet again, I don't see how this follows.
Any help would be greatly appreciated.
$\DeclareMathOperator{\Hom}{Hom}\newcommand{\cat}{\mathcal}$ You have made a subtle mistake in your write-up. The structure of $G$ as a cocone over the above diagram is NOT given by natural transformations $\Hom_{\cat C}(A,-)\overset{\psi_A}\Rightarrow G$ such that $\psi_A=f^*\circ\psi_B$ for each morphism $A\xrightarrow{f}B$ in $\mathcal C$. It is given by natural transformations $\Hom_{\cat C}(A,-)\overset{\psi_{(A,x)}}\Rightarrow G$ such that $\psi_{(A,x)}=f^*\psi_{(B,Ff(x))}$ for each $A\xrightarrow{f}B$.
In other words, you mistakenly indexed the structure maps of the cocone not by the objects of the category of elements, but by the objects of the image of that category under the diagram functor.
With that correction in place, what you want to do is show that there exists a unique natural transformation $F\overset\eta\Rightarrow G$ such that each structure map $\Hom_{\cat C}(A,-)\overset{\psi_{(A,x)}}\Rightarrow G$ indexed by $x\in FA$ factors as $\Hom_{\cat C}(A,-)\overset{\phi_{(A,x)}}\Rightarrow F\overset{\eta}\Rightarrow G$ where $\Hom_{\cat C}(A,-)\overset{\phi_{(A,x)}}\Rightarrow F$ is the natural transformation corresponding to $x\in FA$ by the Yoneda lemma. Consequently, the fact that we want the equations $\psi_{(A,x)}=\eta\circ\phi_{(A,x)}$ forces our definition of $F\overset{\eta}\Rightarrow G$ as follows.
Suppose we wanted to determine where $x\in FA$ gets sent to by the component $FA\xrightarrow{\eta_A}GA$. Well $x\in FA$ is the image of $\mathrm{id}_A$ under the component $\Hom_{\cat C}(A,A)\xrightarrow{\psi_{(A,x),A}}FA$ of the natural transformation $\Hom_{\cat C}(A,-)\overset{\psi_{(A,x)}}\Rightarrow FA$ (that's how the Yonda lemma works under the hood). Hence, we see that $\eta_A(x)$ is the image of $\mathrm{id}_A\in\Hom_{\cat C}(A,A)$ under the component at $A$ of $\eta\circ\phi_{(A,x)}$. But as observed above we want $\eta\circ\phi_{(A,x)}=\psi_{(A,x)}$, hence $\eta(x)$ should also be the image of $\mathrm{id}_A\in\Hom_{\cat C}(A,A)$ under the component at $A$ of $\psi_{(A,x)}$. But again by the Yoneda lemma that is precisely the element $z\in GA$ corresponding to $\Hom_{\cat C}(A,-)\overset{\psi_{(A,x)}}\Rightarrow G$.
In summary, we have determined that the natural transformation $\eta$, should it exist, must be given by $\eta_A(x)=z$ where $z\in GA$ corresponds to $\Hom_{\cat C}(A,-)\overset{\psi_{(A,x)}}\Rightarrow G$. Note that this determines $\eta$ as a ``transformation'', so the only way in which it could fail to exist as a natural transformation would be if it failed to be satisfy naturality.