Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help.

Is it possible for two categories to satisfy two different set-axiom system. Namely- is it possible for two distinct categories $SetZFC$ and $SetZF\neg C$ to exists? One satisfying AC and the other negating it?

And finally, and more generally- Are categories subjected to any a-priory set of set-axioms? or are those axioms given as information in a given category?

Thanks

The question seems to me can you describe the difference in the language of categories? In $Set(ZFC)$ every epimorphism splits, in $Set(ZF\neg C)$ not.

Yes, this is modelable in topos theory. The axiom of choice for toposes says that the any infinite product of non-initial objects is non-initial (one way to state it) or that every epi splits (as noted above). Look up Bill Lawvere's ETCS, which has a choicified version and an non-choicified version.