It is stated on the [nlab](https://ncatlab.org/nlab/show/skeleton#equivalents_of_choice) that the axiom of choice is equivalent to the statement that all small categories have a weak skeleton, meaning a skeletal category which is equivalent to them.



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> Is the [axiom of global choice](https://ncatlab.org/nlab/show/axiom+of+choice#Variants) in MK equivalent to the statement that all (possibly large) categories have weak skeletons?
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Global choice and skeletons of large categories
The answer is yes, by the same argument as for small categories and regular choice. First, assuming global choice, you can form skeleta by picking out one element from each isomorphism class of objects.


EDIT: As Sergei Akbarov points out in the comment, the argument is not quite right - to apply global choice we need to apply it to a class of sets, but the collection of isomorphism classes need not consist of just sets (in fact most natural categories don't have this property). One way to salvage this argument is to exploit [Scott's trick](https://en.wikipedia.org/wiki/Scott%27s_trick): replace each isomorphism class $I$, which may be a proper class, with a set $I\cap V\_\alpha$, where $V\_\alpha$ is the least level of the [cumulative hierarchy](https://en.wikipedia.org/wiki/Von_Neumann_universe) whose intersection with $I$ is nonempty. Thus we have replaced each isomorphism class with its nonempty subset, and we can apply global choice to the resulting class of sets.




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Conversely, suppose we have a class $F$ of nonempty sets and we want to find a choice function for it. Take a category $C$ whose objects are the disjoint union of elements of $F$, and we have exactly one morphism between any two objects belonging to one element of $F$. Then the sets of $F$ are the isomorphism classes from $C$, and any (weak) skeleton constitutes a choice function for $F$.


