Worst category with first isomorphism?

Question

I am no expert in category theory, but from VIII of Algebra: Chapter 0 I learnt that

In an abelian category every $A\xrightarrow{\phi}B$ can be decomposed into \begin{equation}A\twoheadrightarrow C\cong D\rightarrowtail B. \end{equation}

So in abelian categories, every morphism can be decomposed into a epimorphism, followed by an isomorphism, and then an monomorphism, which is the so-called first isomorphism theorem in some famous categories $\operatorname{Set}$, $\operatorname{Grp}$, $\operatorname{Ring}$ and $R\operatorname{-Mod}$.

However, we notice that $\operatorname{Ring}$ is not actually abelian modules and $\operatorname{Grp}$ is not even additive. So the axioms for abelian categories are not actually necessary for the first isomorphism.

I am asking whether there is a natural minimum requirement for a category to have the first isomorphism. If there is no such requirement, then maybe someone can share some bad categories in which one still has the first isomorphism.

Thanks very much!

Answer 1

The first isomorphism theorem says more than this (e.g. it says something about kernels), but if all you want is the existence (and not necessarily uniqueness) of epi-mono factorizations, then any regular category has this property. This includes categories of models of finitary algebraic theories.

If you want uniqueness, note that uniqueness actually fails for $\text{Ring}$...