Let $s':P\to T$ and $t':P\to S$ be two morphisms in an abelian category. We consider their pushout square and factor the horizontal sides using the first isomorphism theorem:
where the middle vertical arrow is induced from the universal property of kernels (using that $\operatorname{im} s=\ker(\operatorname{coker} s)$), and the left square commutes due to the fact that $\operatorname{im}s$ is monic.
Is the left square above also a pushout? If so, why?
Best Answer
$\require{AMScd}$I'll try my best to use your notation.
Consider the span
$$\begin{CD} P @>s'>> T \\ @Vt'VV @. \\ S \end{CD}$$
factored as follows:
$$\begin{CD} P @>c(s')>> I' @>i(s')>> T\\ @Vt'VV @. \\ S \end{CD}$$
where $c(-) = \text{coim}$ and $i(-)=\text{im}$. Now form the pushouts
$$\begin{CD} P @>c(s')>> I' @>i(s')>> T\\ @Vt'VV @VVV @VVV \\ S @>>u> Q @>>v> M \end{CD}$$
Your question is equivalent to: "is $Q$ isomorphic to $I$, and the factorization $S \to I \to M$ isomorphic to $S\to Q \to M$?"
The answer seems to be yes: first of all notice that the universal property of the pushout $Q$ yields a unique map $Q\to I$; we aim to show that this map is invertible.
Observe that the factorization $P\to I'\to T$ of $s'$ is just the epi-mono factorization of $s'$, and epimorphisms are stable under pushout; so $u$ is an epimorphism.
If we show that $v$ is a monomorphism, we have an epi-mono factorization of $S\to M$, that must be isomorphic to the epi-mono factorization $A \xrightarrow{c(s)} I \xrightarrow{i(s)} M$ of $s$, because such a factorization is unique up to isomorphism (if you have two $(\cal E,M)$-factorizations of the same map, use orthogonality to show that there is an isomorphism between the factoring objects).
But in an abelian category, monomorphisms are also stable under pushout, so we're done: since $i(s')$ is a mono, $v$ is a mono. So, we're done. $\square$
This is not relevant to your question, but the two squares obtained in this fashion are not only pushouts, but also pullbacks (lacking a consensus on the terminology, I called them "pullouts" when I had to study them, A.2.3 here): this is the content of the Stacks lemma I mentioned.
Points to keep in mind: the coim/im factorization in an abelian category $\cal A$ corresponds to the epi-mono factorization; this factorization is pushout and pullback stable, because pushouts where a leg is monic are also pullbacks, and pullbacks where a leg is epic are also pushouts; a distinguishing feature of abelianity is this self-duality (pullback iff pushout is the way in which $\infty$-/homotopy coherent notions of abelianness are stated); this reduction is very powerful: for example, it proves in a one-liner the fact that the heart of a t-structure is abelian.