Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and $\mathcal{Y}$ are conglomerates and $f: \mathcal{X} \rightarrow \mathcal{Y}$ is a surjective map, then $f$ has a right inverse.
The "Axiom of Choice for Classes" and the (usual) "Axiom of Choice for Sets" may be defined in the obvious, analogous way (i.e., "Whenever $\mathcal{X}$ and $\mathcal{Y}$ are classes $\ldots$ etc." in the first case, and "Whenever $X$ and $Y$ are sets $\ldots$ etc." in the second).The Axiom of Choice for Classes implies the existence of a global choice function (meaning, a class function which is a choice function for the class of all non-empty sets), and the latter is, in turn, equivalent to the existence of a well-ordering of the universe.
Well, it is well-known that the Axiom of Choice for Conglomerates implies that "Every category has a skeleton" (see, e.g., Adamék/Herrlich/Strecker book). On the other hand, Isbell and Wright have proved in the 60's that the statement "Every category has a skeleton" implies the existence of a well-ordering of the universe.
My question is: considering the following assertions,
"The Axiom of Choice for Conglomerates"
and
"Every category has a skeleton"
are they equivalent statements ? At first glance, my conjecture is that the answer is "Yes", but I didn't find any reference for that.
Added (1): In Freyd/Scedrov book it is shown that the Axiom of Choice for Sets is equivalent to the statement "Every small category has a skeleton". So I also wonder whether there is some ladder of equivalences between forms of the Axiom of Choice and assertions regarding the existence of skeletons for certain categories. I mean, considering statements as
"Every small category has a skeleton",
"Every locally small category has a skeleton", and
"Every category has a skeleton", and maybe other similar statements of this kind,
could we put each one of them in correspondence with some equivalent form of the Axiom of Choice ? The first one is equivalent to the Axiom of Choice for Sets, as I have just commented.
Added (2): After getting some comments it seems that the set-theoretic background should be more specified. The question was posed assuming the usual foundations of category theory - but, of course, this is a very debatable issue. In a first moment, we could think of ZFC + 2 inaccessibles, and then in such environment code the notions of: set, class and conglomerate. However, whether we are in first or second order seems also to have influence on this matter.
I was also told that, indeed, the Axiom of Choice for Classes is equivalent to the Axiom of Choice for Conglomerates: the Axiom of Choice for Classes, as stated above, is equivalent to the existence of a well-ordering in any class, and so we could use such well-ordering to define a right inverse.
So, I guess that, indeed, my real question is: how can we define precisely a background (both categorically and set-theoretically) where we could analyze statements as the ones listed at the end of the question,
"Every small category has a skeleton",
"Every locally small category has a skeleton",
"Every category has a skeleton", etc.,
and what forms of the Axiom of Choice would these statements be equivalent to ?
