As many people here will be aware, there is a debate within the foundations of category theory as to how to approach the discipline. One can either:
Axiomatise the category of all categories directly using one's favourite axiomatisation.
Interpret the categories set theoretically within an inaccessible cardinal. In this way we can mimic reasoning about the proper-class like entities of category theory with sets in the inaccessible model.
(A third method that deserves mention is to introduce a class theory (say $VNBG$ or $MK$) and do the interpretation there, but I'm not too concerned with this)
An obvious shortcoming of method (2) is that it is hard to talk about functor categories between large categories, as even one functor is already not an object in the inaccessible domain, so sets of these functors do not exist in the domain.
Method (2.) was then extended by Grothendieck to deal with this problem by proposing the following axiom:
[Grothendieck's Axiom] Every set $x$ is a member of some universe (In particular; for any universe $V$ there is a larger universe $V'$ such that $V \in V'$).
We can then interpret, say, functor categories between large categories of a given set theoretic model $V$ as sets within an extended model $V'$.
However, Grothendieck's method seems to have a serious shortcoming; it cannot account for the category of all categories (henceforth denoted by Cat) . For such a category does not occur as an object in any of these universes; there will always be objects and categories outside of any particular universe.
I am interested in exactly how much is at stake here. My question then is twofold:
Is there a way of using a Grothendieck paraphrase to dodge this problem and nonetheless at least mimic talk of a category of all categories?
Are there places in the literature where category theorists and other mathematicians make serious use of a category of all categories as an object of the theory (e.g. by showing that there is a functor from Cat to a different category)? Particularly welcome would be examples from applications of category theory to non-category theoretic mathematics (such as in the proof of Fermat's Last Theorem or in Grothendieck's work).