TITLE: Equivalence of categories and axiom of choice
QUESTION [7 upvotes]: I don't understand much about set theory but nevertheless try to learn category theory. In books like Gelfand Manin, there is the well known theorem that a functor defines an equivalence of categories iff it is fully faithful and essentially surjective. The proof implicitely uses some kind of axiom of choice for proper classes, which, I guess, cannot be proved in ZFC.
From my mathematical education I am used to only use ZFC for proving theorems and even explicitely mention every use of the axiom of choice for sets. So from my mathematical education I would say, that the above theorem is simply "wrong" and should not be used by a any serious mathematician working outside of set theory, e.g. number theory or algebraic geometry.
As this theorem is stated in every place without comment (okay, Wikipedia has a very sloppy comment, which doesn't state which kind of axiom of choice in what kind of strange set theory is needed), I wonder if all the category theorists have as less clue about set theory than me and what parts of pure mathematics are "wrong" because of nobody cares? Or is there some good set theoretic argument in favour of this theorem.

REPLY [4 votes]: The choice-free statement of the cited theorem is as follows:

If $F : \mathcal{C} \to \mathcal{D}$ is fully faithful and constructively essentially surjective on objects, i.e. there exist a map $G : \operatorname{ob} \mathcal{D} \to \operatorname{ob} \mathcal{C}$ and a map $\epsilon : \operatorname{ob} \mathcal{D} \to \operatorname{mor} \mathcal{D}$ such that each $\epsilon (D)$ is an isomorphism $F (G (D)) \to D$, then $F : \mathcal{C} \to \mathcal{D}$ is (half of) an equivalence of categories.

In the absence of choice, the hypotheses are strictly stronger, but they are often satisfied in practice – for example, $F$ might be the restriction of an adjunction, or something like that. There are similar problems with the definition of "complete category" etc. The trouble with giving constructively correct versions of all these general definitions and theorems is that it would just obscure the essential idea without much payoff. Thus, for clarity and convenience, we assume the axiom of (global) choice, universes, etc.
Incidentally, there is also a way to fix the naïve version of the cited theorem by changing the definition of "functor" – specifically, we could work with anafunctors instead of functors. This is related to the idea of formally inverting weak equivalences. I think you will agree that it is not at all helpful to introduce anafunctors to beginners in category theory.