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?