On some diagram assertions in preabelian and $P$-semi-abelian categories

As is well known, many important additive categories in functional analysis and algebra are not abelian. Many classical diagram assertions valid in abelian categories fail in more general additive categories without additional assumptions concerning the properties of the morphisms of the diagrams under consideration. This in particular applies to the so-called Snake Lemma, or the Ker-Coker-sequence. We obtain a theorem about a diagram generalizing the classical situation of the Snake Lemma in the context of categories semi-abelian in the sense of Palamodov. It is also known that, already in $P$-semi-abelian categories, not all kernels (respectively, cokernels) are semi-stable, that is, stable under pushouts (respectively, pullbacks). We prove a proposition showing how non-semi-stable kernels and cokernels can arise in general preabelian categories.