Translating a diagram chase into an element-free proof

$\newcommand\img{\mathrm{img}}$

It's a bit more involved. I needed the fact that, in an Abelian category, pullbacks preserve epimorphisms. If you are willing to keep going, the next step would be to see that, since $(\ker n\to C \xrightarrow h D)=0$, then $\ker n\to C$ factors through $\ker h\to C$, which by exactness is the same as $\img g\to C$. Then take the pullback of $\ker n\to\img g$ and $B\to\img g$... this is the 5th step in your proof. Can you continue?

Thanks to the answer and the comment by Jackozee Hakkiuz, I did the following proof. (The notations are not exactly the same, but it should be clear. Also the proposition 1.3.3 is simply the fact that the pullback of an epimorphism is epic.)

I hope that this may be useful to someone :)