A general version of the 5 lemma

Question

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 $f,h\in\cal M$ $\Rightarrow g\in\cal M$.

1. I'm not assuming that $(\cal E,M)$ is proper (so there can be no relation between $(\cal E,M)$ and $(Epi, Mono)$);
2. 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.

Answer 1

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! (-: