I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}\_F$. In many articles one introduces the set $\hat{\mathcal{W}}\_F$ of all equivalence classes of irreducible representations of $\mathcal W\_F$. 


It seems to me that strictly speaking this does not exist (due to set-theoretical reasons). Instead one thinks of $\hat{\mathcal{W}}\_F$ as a set that contains for every irreducible representation of $\mathcal{W}\_F$ an isomorphic copy of it. So elements of $\hat{\mathcal{W}}\_F$ are true representations (and not some collection of representations). In order to create $\hat{\mathcal{W}}\_F$ we need to be able to make a universal choice to pick a 'canonical model' from every class of isomorphic irreducible representations.


My questions are the following: 


* Which axioms people assume in order to define a true set like $\hat{\mathcal{W}}\_F$?
* Does it following from the axioms used for example in Bourbaki? (to answer this one needs to be familiar with their foundations as described in 'Theory of Sets'.


Other considerations/remarks related to my question are also welcome. Thanks in advance.



"set of all irreducible representations of a group", set-theoretic issues
It is easy to show that the cardinality of the underlying set of any irreducible representation is bounded in terms of the cardinality of $\mathcal{W}\_F$, so can you just take your favorite set $S$ of sufficiently large cardinality and consider only irreducible representations whose underlying set is a subset of $S$. More generally, whenever you can get a cardinality bound on some kind of object, you can use this to get a set of such objects that represents every isomorphism class.


