There are some basic discussions on the motivations of large categories and small categories [here] (On the large cardinals foundations of categories) and [here] (Large cardinal axioms and Grothendieck universes) and [here] (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 an discussion [here] (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 !