Non-pointed abelian categories

We study a property (P) of pushouts of regular epimorphisms along monomorphisms in a regular context. We prove that (P) characterizes abelian categories among homological ones. In the non-pointed case, we show that (P) implies the normality (in the sense of Bourn) of all subobjects, that any protomodular category satisfying (P) is naturally Mal’tsev, and that an exact category is penessentially affine if and only if it is protomodular and satisfies (P). An example of such a category is the one whose objects are the abelian extensions over an object in a (strongly) semi-abelian category; by exploiting some observations in this context, we also provide a characterization of strongly semi-abelian categories by a variant of the axiom of normality of unions.