Is the axiom of choice an assumption, that one may "freely" choose (eg, ZFC) or discard (eg, ZF, ZF+AD), or is it determined by the nature of the categories being considered?
The latter view is expoused in Lawevere & Rosebrugh's Sets for Mathematics where it's stated that Choice is false in categories with "internal motion and cohesion", as opposed to, eg, the category of constant sets where Choice is true.