It is frustrating to hear people speak of Yoneda embedding, category of all categories/functors, n-categories, infinity categories and all that jargon, without giving proper logical justifications.

I learned category theory from N. Jacobson, Basic Algebra - II. The justification given therein, that one uses the Godel-Bernays distinction of sets and classes, simply does not work for the above cases.

This is really frustrating. How do people deal with it? It seems many times it is skipped simply, giving the impression that it is too unimportant to be dealt with.

How did then the more foundational guys, for instance, Grothendieck deal with it? What are the "universes" one hears from time to time?