semi-abelian category
pre-abelian category in which the induced morphism coim 𝑓 → im 𝑓 is a bimorphism for any morphism 𝑓
quasi-abelian category
pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and the pullback of a cokernel along arbitrary morphisms is again a cokernel