On the axiomitization of an optimal noetherian form over the category of sets

A noetherian form is an abstract self-dual framework suitable for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for non-abelian, as well as abelian, group-like structures. It can be seen as a unification of semi-abelian categories and Grandis exact categories (two separate generalisations of abelian categories) that retains the self-dual character of Grandis exact categories. In this paper we axiomatise a noetherian form over the category of sets, which, in a suitable sense, is an optimal noetherian form. We then show that every topos has such noetherian form as well, while in the context of a pointed category with finite products and sums, the dual of such noetherian form is exactly the form of subobjects of a semi-abelian category.