3×3 Lemma and Protomodularity

The classical 3 × 3 lemma and snake lemma, valid in any abelian category, still hold in any quasi-pointed (the map 0 → 1 is a mono), regular, and protomodular category. Some applications are given, in this abstract context, concerning the denormalization of kernel maps and the normalization of internals groupoids (i.e., associated crossed modules).