Axioms for category theory

The standard axioms vary: they're either ZFC with an axiom of choice for proper classes, some set theory such as NBG that axiomatizes classes more thoroughly, or ZFC with Grothendieck universes, so that "large" categories are interpreted as still being small, but relative to a larger "universe" of sets. There have been efforts to axiomatize category theory without set theory, most notably ETCC, the elementary theory of the category of categories, but these have not proven to be sufficient as a foundation.