In some books on category theory (for example, in J.Adámek, H.Herrlich, E.Strecker "Abstract and concrete categories...") the authors use the idea of "big sets" ("conglomerates" or "collections") which can contain classes (as far as I understand, in the Goedel-Bernays sense) as elements, and they formulate the "generalized axiom of choice", where it is stated that the choice function exists (not only for families or classes of sets, but also) for families of classes (indexed by elements of those "big sets"). This approach allows to prove, in particular, the existence of a skeleton in each category, and some other useful things.
This generalization of the axiom of choice is also mentioned In Wikipedia: http://en.wikipedia.org/wiki/Axiom_of_global_choice (as the "strong form of the axiom of global choice").
I wonder if there are any texts with the justification of this trick? The references I found (in particular, those mentioned in Wikipedia) give justification only for usual axiom of choice (for families of sets or for classes of sets, but not for "conglomerates of classes"). So actually I can't understand whether, for example, the existence of a skeleton, is true for all categories (in some interpretation of set theory) or for some special ones... Similarly the other corollaries of this "global axiom of choice" look doubtful for me. I would be grateful if anybody could clarify this.
From the comments I see that there is a risk of misunderstanding, so I want to explain that by justification I mean an accurate (rigorous) definition of the new tool together with the analysis of whether it is compatible with the other tools of the theory.
As an illustration, in the case of the usual axiom of choice (I mean its "weak form", in terms of Wikipedia), there are many textbooks (I can recommend E.Mendelson "Introduction to mathematical logic" or J.Kelly "General topology", the appendix), where the fundamental objects of the theory (in this case, the classes) are accurately introduced (here, axiomatically) and the necessary constructions (like functions) are rigorously defined in the theory. This makes possible to give rigorous formulation to the axiom of choice (again, to its "weak form") inside the theory, and moreover, this presentation of a new axiom is followed by a thorough investigation of whether it contradicts to the previous axioms of the theory. Only after receiving the answer that no contradictions can appear (in fact, a more strong thing is true: the new axiom is independent from the others, that was the result by P.Cohen) mathematicians can use this new axiom without worrying that something is wrong here.
So my question is whether there is something similar for the "strong form of the axiom of choice"? Is it possible that nothing lies behind these words? On the contrary, if there is a justification, where can I read about it?
Dear colleagues, from what I learn on this subject in the textbooks which I found, in Wikipedia and here in MO, I deduce that what people call "axiom of global choice" is just the usual axiom of choice (as it is presented in Kelly's book) applied to some special classes of sets arising in consideration of what is called the Grothendieck Universe. It's a puzzle for me
1) why people call this special case "a stronger form of the axiom of choice", and
2) why they don't want to give references, where this construction is accurately introduced.
With the aim to accelerate the clarification of this question, I now nominated for deletion the article in Wikipedia devoted to his topic: http://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Axiom_of_global_choice. As I wrote there, I don't exclude that the partisans of the idea will rewrite the article in Wikipedia for endowing "global choice" with some mathematical sense, but you should agree that in its present form this article and the other mentionings of "global choice" available for external observers, look indecently vague. I invite all comers to share their opinion here or at the Wikipedia page.