So if you are willing to go full Grothendieck from the get-go you can dodge the need to have algebraic closures. Even making an algebraic closure doesn't require the axiom of choice, as you can adjoin the roots of all polynomials with coefficients in your field without invoking it.
The problem is that a lot of categorical constructions involve sets that get big. This isn't too much of a problem, but if you want to show that they are nonempty, choice comes in in a big way. Even worse, ZFC isn't enough: you need to play tricks to formalize category theory inside of set theory (or use proper classes instead).
The problem is that a lot of categorical constructions involve sets that get big. This isn't too much of a problem, but if you want to show that they are nonempty, choice comes in in a big way. Even worse, ZFC isn't enough: you need to play tricks to formalize category theory inside of set theory (or use proper classes instead).