I am struggling with exercise VIII.4.2 in Mac Lane's Categories for the working mathematician. The exercise is about proving the four lemma about being an epimorphism using members in abelian categories (Borceux calls these pseudo-elements).
I am trying to adapt the proof in the category of abelian groups to use members, but I am getting stuck near the end.
My proof so far goes like this:
Let $x \in_m b_3$. Since $f_4$ is an epimorphism, by Theorem $3(iii)$ we find $y \in_m a_4$ such that $f_4y \equiv h_3x$. Since $f_5$ is a monomorphism, $g_4y\equiv 0$ and by Theorem $3(v)$ we find $z \in_m a_3$ satisfying $g_3z \equiv y$. Now $h_3f_3z \equiv f_4g_3z \equiv f_4y \equiv h_3x$, so by Theorem $3(vi)$ we find some $a \in_m b_3$ satisfying $h_3a\equiv 0$ (among other things). By Theorem $3(v)$, this gives $b \in_m b_2$ such that $h_2b \equiv a$, and by 3(iii) we have a $c \in_m a_2$ satisfying $f_2c \equiv b$.
Now in the proof for abelian groups, we could show that $f_3(g_2c + z) = x$ (maybe up to sign, depending on how exactly $a$ was defined), but I don't see how to apply Theorem $3(vi)$ in the correct way to do this.
Here are the definition of members and Theorem 3:
Here is the statement of the Five Lemma:
And here is the exercise in question:
