I'm trying to find some insights about cokernel pairs. I understand the definition and I know how to calculate them in concrete categories, but while the kernel pairs seem to have a very sensible interpretation and appear in the literature a lot, I'm curious wether cokernel pairs also are somewhat meaningful. Since in case of proper categories we can identify the coequaliser of the kernel pair with image of $f$, probably as everything else in the abstract nonsense circle must have a dual statement that equalisers of cokernel pairs correspond to coimages in case of working with well-behaved categories? However, since the notion of the coimage is much more meaningless to me than the initial definition by a pushout, that it's not very helpful, even if true.
For those not familiar with the object, the cokernel pair of a morphism $f$ is the colimit of the diagram $\require{AMScd}$ \begin{CD} X @>>{f}> Y \\ @VV{f}V &\\Y \end{CD}
Which is just the pushout over duplicated morphism $f$.
I can offer you the following insight:
Lemma: $f\colon X\rightarrow Y$ is epic iff the universal morphism $Y+_XY\rightarrow Y$ is an isomorphism.
Proof:
$\Rightarrow$: Since epimorphisms are stable under pushout, the morphism $Y\rightarrow Y+_XY$ is epic. Since it is also a section of the universal morphism $Y+_XY\rightarrow Y$ in the left diagram, it is an isomorphism. Therefore $Y+_XY\rightarrow Y$ is an isomorphism as well.
$\Leftarrow$: Let $g_1,g_2\colon Y\rightarrow Z$ be morphisms with $g_1\circ f=g_2\circ f$. Since $Y$ is a pushout of $Y\xleftarrow{f}X\xrightarrow{f}Y$, we get $g_1=g_2$ as the universal morphism in the right diagram. Therefore $f$ is epic. $\square$