In an additive category, given a morphism $f : A\to B$ let $\operatorname{im} f = \ker(\operatorname{coker}(f))$ and $\operatorname{coim} f = \operatorname{coker}(\ker(f))$, if they exist. If they do, then there is a unique morphism $\tilde f$ such that $f = \operatorname{im} f \circ \tilde f \circ \operatorname{coim} f$.

An additive category is "*semi-Abelian*", if every morphism a has a kernel and cokernel and the $\tilde f$ is mono and epi for all $f$.

I put 'semi-Abelian' in quotation marks because this definition conflicts with the on the nLab, but this definition is given in "Derived Functors in Functional Analysis" by J. Wengenroth (although I don't know who introduced the term originally).

> Are "semi-Abelian" categories regular? If not do they have regular
> epi-mono factorizations?


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This might be "obvious", but it isn't to me. Also, I'm also interested in the question with "regular" replaced by "protomodular", but I'm going to postpone that question for now.