In abelian categories for a morphism $u:X\to Y$ the cannonical morphism $\bar{u}:\mathrm{Coim}(u)→\mathrm{Im}(u)$ is an isomorphism. However this is a property, not a part of the axioms of abelian categories. There are different weaker types of categories (pre-, semi- and quasi-abelian) that allow this situation (I believe, initially quasi-abelian categories were called semi-abelian, f.e. in papers of Raikov, and now this term is used differently, but I am not sure by now).
My question is - is there a unified system of axioms for those categories? So that one could say that, for example, Preabelian+(Axiom A)=quasi-abelian, quasi-abelian+(Axiom B)=abelian, etc.
