  We show that non-pointed versions of the classical homological lemmas hold in
regular protomodular categories equipped with a suitable posetal
monocoreflective subcategory. Examples of such categories are all protomodular
varieties of universal algebras having more than one constant, like the ones of
unitary rings, Boolean algebras, Heyting algebras and MV-algebras, their
topological models, and the dual category of every elementary topos.
