pull back and push out problem could you please help with the proof?
If a category $K$ has pullbacks.  For composable morphisms $f: A\to B$ and $g: B \to  C,$ if $g$ and $f$ are extremal epimorphisms, prove that $gf$ is an extremal epimorphism.
The definition of extremal epimorphism I have been given is:
A morphism $e: A \to B$ is an extremal epimorphism when for each commutative diagram $e: A \to B,$  $f: A \to C$, $m: C \to B$ ($e= mf$) if $m$ is monic, then $m$ is an isomorphism.
I have let $f = mk$ and $g = nh$,  then I have drawn out the pullbacks of $f$ and $m,$ and of $g$ and $n$.
I don't know where to go from here.

Let $h:A\to X$ be any morphism and $m:X\to C$ monic such that $mh=gf$. We want to show that $m$ is an isomorphism.
Let $P$ be the pullback of the diagram $g:B\to C$ and $m:X\to C$. By the universal property of pullbacks there exists a map $\alpha:A\to P$. Denote by $m':P\to B$ and $g':P\to X$ the induced morphisms. Now $m'$ is monic as the pullback of a monic morphism.
Because $f$ is an extremal epi, $m'$ is an iso. Hence we have a morphism $g'm'^{-1}:B\to X$. Because $g$ is an extremal epi, $m$ is an iso.
Edit: The fact that $m'$ is monic is not entirely trivial. Maybe you want to show this yourself.
