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? Usually by cheating, I suppose. How did the serious guys, for instance, Grothendieck deal with it? What are the "universes" one hears from time to time?