Consider the category of epimorphisms $\mathcal E$ in a given abelian category, where epimorphisms are objects, and morphisms of this category are pairs of arrows which make its objects commute. That is, let $f_1,f_2\in Ob(\mathcal E)$ be epimorphisms and $\alpha:(g,h):f_1\to f_2$ in $Hom(\mathcal E)$ be such a morphism which makes the following commute:
$\newcommand{\la}[1]{\kern-1.5ex\xleftarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} D & \la{f_1} & 0 \\ \da{g} & & \da{h}\\ 0 & \la{f_2} & 0 \\ \end{array}$
Is it possible to deduce when $\alpha$ is a epimorphism (or monomorphism)? For example, since $f_1,f_2$ are both epimorphisms, can one show that $\alpha$ is an epimorphism $\iff$ $h$ is an epimorphism?