so let's try to explicitly describe the syntactic category of "the algebraico-geometric theory of one epimorphism"...

what about something like "homological algebra" and "derived functors" and so forth for finitely cocomplete algebroids that are far from being abelian?? ...or something like that...

also, consider the forgetful algebro-geometric morphism from the moduli stack of epimorphisms to the moduli stack of morphisms ...