I'm reading this proof of the Five Lemma (I only included the injectivity part here) and got stuck in the very last step:
What I don't understand is how the author concludes that $x=\beta \circ \alpha (z)=0$. I can't see the connection to this statement from the preceding step right now (I can follow everything else until that point). It's probably something obvious but I'm stuck.
(By the way, I think there's a typo when it says that $b(y)=b\circ \alpha (y)$, it should probably be $b(y)=b\circ \alpha (z)$).
Since $b(y) = b\circ \alpha(z)$ and $b$ is injective, $y = \alpha(z)$.
Then $x = \beta(y) = (\beta\circ\alpha)(z) = 0$.