There are some basic discussions on the motivations of large categories and small categories: On the large cardinals foundations of categories, Large cardinal axioms and Grothendieck universes, Small model categories?.
As Mac Lane noted on large categories in categories for working mathematicians:
- One universe assumption is sufficient for all small sets and all small groups but does not provide category for all sets and all groups.
- Grothendieck gave a stronger assumption that for each universe a category of all those groups which are members of the universe. However it does not provide any category of all groups.
- Some proposals defined a category with a set-free terms such as axioms as first order axioms for the category of all sets. This includes elementary topos or logic tools.
- There are some ideas in the above discussions.
- For higher category theory, there is a discussion: Are grothendieck universes enough for the foundations of category theory?.
The two devices of the universes do not address the issue completely. Is the issue still open? References will be very appreciated!