What appears as an obstacle to me is the axiom of regularity, which solves some paradoxical foundational questions and the related well-foundedness concepts: In a category theoretical set theory, e.g. ETCS, the point is to avoid a membership predicate "$\in$" and nestings of sets like $b\in a,\ c\in b,\ d\in c,\dots$. Subobjects are characterized via arrows and the "regular" topoi which are like set theories characterize elements as single injections too.
Does category theory let us formulate any statement which relates to regularity?
Naively it would appear that the theory doesn't leave us any choice but exclude the phenomenon which the classical formulations axiomize away. But then to what extend does the theory leave you a choice and to what extend can you deviate from it?
Appearently non-well-foundedness is a thing now. (And I even read somewhere on MathOverflow that the proof of the abc-conjecture relates to it.) And so I assume you can speak about it in category theory as well.