The fact that any functors which is fully faithfull and essentially surjective is an equivalence of category is equivalent to the axiom of choice.

Without the axiom of choice, Makkai has introduced the notion of 'anafunctor' which generalize the notion of functors and provide inverses for the functors which are essentially surjective and fully faithful.

One can formulate his definition as follow: The category of categories is a Brown category of fibrant objects, with fibrations being the isofibrations and weak equivalences being the fully faithful and essentially surjective functors (trivial fibrations are hence the functors which are fully faithful and surjective on objects).

Anafunctors $X \rightarrow Y$ are then spams $X \leftarrow Z \rightarrow Y$ where $Z \rightarrow X$ is a trivial fibrations, and we know by the work of Brown that homotopy class of such spans compute the localization of the category of categories at 'weak equivalences'.

The problem is that in general there is a proper class of anafunctors between two categories, Makkai proved that under some set theoretical axiom there is a set of equivalence class of anafunctors and hence that this localization is indeed locally small.

My question is: is it known that in only ZF it cannot be proved that this localization is locally small ? i.e. that in ZF there is not always a (small) set parameterizing all equivalences classes of anafunctors between two given categories ?