The axioms for abelian categories are nice and clear. The axioms for protomodular categories - and therefore semi-abelian categories - are beyond me entirely. I'm looking for a breakdown of the definition and a discussion of its components along with some intuition. A reference would be great too. (Bourceux & Bourn's book only gets to protomodular categories on page 227, which is.. not optimal.)
Update: For a nice characterization of semi-abelian (and homological) categories, see Hartl & Loiseau's A working Mathematician's Definition of Semi-Abelian Categories.