TITLE: A general version of the 5 lemma
QUESTION [10 upvotes]: Suppose you have an abelian category $\bf A$, and $A\to B\to C$, $A'\to B'\to C'$ two exact sequences, in a diagram
$$
\begin{array}{cccccccc}
0 &\to & A &\to& B &\to& C &\to &  0\\
&&\downarrow && \downarrow  && \downarrow \\
0 &\to &  A' &\to& B' &\to& C' &\to &  0
\end{array}
$$
(i.e., suppose you have a morphism of exact sequences $(f,g,h)$); let $(\cal E,M)$ be a factorization system on $\bf A$, and suppose $f,h$ lie in $\cal E$. 
Under which conditions (on the category, or on the factorization systems) $g\in\cal E$? Same question for $f,h\in\cal M$ $\Rightarrow g\in\cal M$.

I'm not assuming that $(\cal E,M)$ is proper (so there can be no relation between $(\cal E,M)$ and $(Epi, Mono)$);
For the motivating example I have in mind with this question, assuming something more on the factorization system is more natural than assuming it for the category.

REPLY [3 votes]: One condition on the factorization system that should work is that "factorizations preserve exact sequences", i.e. if you have a map between exact sequences and you $(E,M)$-factor it componentwise, then the intermediate objects also form an exact sequence.  If this is true, then you can factor your given map of exact sequences and apply the ordinary 5-lemma, using the fact that $f\in E$ iff the $M$-part of its $(E,M)$-factorization is an isomorphism.
However, this assumption may not be much easier to check in an example than to check your desired conclusion directly!  (-: