Logical problems in category theory 
Possible Duplicate:
Set theory for category theory beginners 

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?
 A: Short answer: category theorists often elide the extra annotations when employing typical ambiguity or universe polymorphism.  Proof theorists demand that these annotations be provided, and study how they behave.
If you want to be pedantic, then you have to annotate all instances of "category of sets" or "category of categories" with the additional word "small".  Then the objects of the category of small sets do not form a small set, and the category of small categories is not a small category.
The next step is to replace "small" with an arbitrary natural number, where the objects of the "category of 0-small sets" form a 1-small set.  Often, when fully annotated, it turns out that a proof will work for any value of "N" (where all the references to Set or Cat in the proof involve "offsets" from that N, such as "(N+3)-small sets").  Proofs which are parametric in this N (or some sequence N,M,... with inequality constraints between them) are called universe-polymorphic proofs, and are quite similar to a phenomenon in Principia Mathematica called typical ambiguity (although PM asserted a staggeringly powerful axiom about typical ambiguity without any sort of formal justification).  You can re-apply these proofs at arbitrary levels in the transfinite hierarchy of universes, and they still hold.
That said, nobody has yet proven that a category of ALL categories (of every "smallness") cannot exist in the way that Russell proved that a set of all sets cannot exist.  However, there is some evidence that you would have to omit certain axioms that might seem to be "obvious" at first glance.
A: I think you are just reading the wrong books.  The most common solution to these problems is indeed Grothendieck universes.  Really, these issues are not that big a deal, not because they are logically unimportant, but because there are well-understood ways of dealing with them, which generally are extremely effective, and so it's not worth saying more than "small", "large", "very large", etc.—to do so would be distracting.  (You can probably find many such informal uses of language in other fields as well.)
A: Every question asked can be divided into two parts: what is known, and what is asked.
I think your question's "what is known" part is by no means universally agreed. It's not frustrating to hear people talk about categories or $\infty$-categories anymore than hear people talk about sets. 
Yes, when you talk about sets, sometimes you can make mistake if you consider "the set of all sets", but if you're using set theory only to teach algebraic topology, you chances of making such a mistake while proving any interesting theorem aren't big. What's the reason the same doesn't apply here?
Therefore, I do not think it's possible to answer your "what is asked" part.
If you're unconvinced, perhaps you could provide logical inconsistencies in Lurie's Higher Topos Theory? (This is what I read for $\infty$-categories)
