Let $\mathcal{C}$ be a category with the following three properties:
1) $\mathcal{C}$ is finitely complete
2) $\mathcal{C}$ has an extremal-epi-mono factorization
3) extremal epimorphisms are stable under pullbacks
Now, consider a morphism $f:A\to B$ in $\mathcal{C}$ and a pullback of $f$ along an arbitrary arrow $g$:
$$\begin{array} PP & \stackrel{f'}{\longrightarrow} & C \\ \downarrow{g'} & & \downarrow{g} \\ A & \stackrel{f}{\longrightarrow} & B \end{array} $$
By property 2), we can write $f$ as $f=m\circ e$, with $e$ an extremal epimorphism and $m$ a monomorphism. The same can be told about $f'$, so $f'=m'\circ e'$. Now consider the diagram $$\begin{array} PP & \stackrel{e'}{\longrightarrow} & I' & \stackrel{m'}{\longrightarrow}&C \\ \downarrow{g'} & & \downarrow{g''} & &\downarrow{g} \\ A & \stackrel{e}{\longrightarrow} & I & \stackrel{m}{\longrightarrow}&B \end{array} $$ Is it true that $(P,e',g')$ is a pullback of $(e,g'')$ and that $(I',m',g'')$ is a pullback of $(m,g)$, for a suitable $g''$?
In other words, is it true that not only extremal epimorphisms, but the whole epi-mono factorization is stable under pullbacks?