Given an $R$-module $M$, and submodules $M_1, M_2 \subseteq M$, we may form the “sum” of the submodules $M_1 + M_2 \subseteq M$, where $M_1 + M_2 = \{ x + y : x \in M_1, y \in M_2 \}$. I would like to carry out a similar construction in an arbitrary Quillen Exact Category.
Suppose that we have an exact category $\mathcal{E}$ with its class of designated short exact sequences $E$. Let $f:A \rightarrow M$ and $g: B \rightarrow M$ be admissible monomorphisms in $\mathcal{E}$. To form $A + B$, I want to first form $A \cap B$ as the pullback:
$\require{AMScd}$ \begin{CD} A \cap B @>{f’}>> B\\ @Vg’VV @VgVV\\ A @>{f}>> M \end{CD}
We then use these new maps $f’$ and $g’$ to take the pushout:
$\require{AMScd}$ \begin{CD} A \cap B @>{f’}>> B\\ @Vg’VV @Vg’’VV\\ A @>{f’’}>> A + B \end{CD}
It is clear that this construction gives the correct result for modules. I have one issue though in a general exact category. While it is clear that $f’$ and $g’$ will be monomorphisms (pullbacks of monos are monos), it is NOT clear to me that these will be admissible monomorphisms. If they are, this construction works because pushouts of admissible monomorphisms are admissible monomorphisms.
Is there a way to see that pullbacks of admissible monomorphisms along admissible monomorphisms are themselves admissible monomorphisms? If this is not true, is there a simple extra condition to put on the category that will make this work?