let me say that I am not a set theorist, but I have to settle up some things in category theory and I need your help. 


What I'd like to do is, in some way, use axiom of choice for proper classes.
I say that axiom of choice holds for a set $X$ if there exist a function $f:X \to \bigcup X$ such that $f(x) \in x$.


My attempt is the following:


Assume the Tarski Grothendieck axiom, i.e. every set is contained in a Grothendieck Universe.


Fix a Universe $U$ and an enlarged universe $U^+$ that contains $U$.
Now note that every subset of $U$, i.e. a $U$ proper class, is contained in $U^+$ by the subset axiom. Now I read from wikipedia that "axiom of choice holds" in Tarski Grothendieck framework. My question is:



> 
> Does the axiom of choice holds in TG for every element of some universe?
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> 
> 


This would mean that for every subset of $U$, axiom of choice holds, being an element of $U^+$. To be honest, I would be ok with the following:



> 
> Does the axiom of choice holds for a Universe? Can you provide a reference?
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> 
> 



Very large axiom of choice
If you have ZFC in the ambient theory, including the axiom of choice, then indeed the axiom of choice holds in every Grothendieck-Zermelo universe (also sometimes known as Grothendieck universes). A Grothendieck-Zermelo universe is a rank-initial segment $V\_\kappa$ of the cumulative hierarchy, where $\kappa$ is an inaccessible cardinal. And every set in $V\_\kappa$ has a well-ordering in $V$, by the axiom of choice in the ambient theory, and this order must also be in $V\_\kappa$, since $V\_\kappa$ is closed under subsets of its elements. 


Basic lesson: if the axiom of choice holds in the background theory, then it also holds in every Grothendieck-Zermelo universe. 


Meanwhile, it is not a consequence of the Universe axiom over ZF that AC must hold, since one can easily use forcing to make AC false, while preserving the truth of the universe axiom. So without AC in the ambient theory, you cannot conclude that AC holds in every Grothendieck-Zermelo universe.


