The Hierarchy of “collections” for Categories

Question

When we deal with usual category $\mathbf A$ we require that Ob$(\mathbf A)$ (i.e., collection of objects of category $\mathbf A$) and Mor$(\mathbf A)$ (i.e., collection of morphisms between objects in category $\mathbf A$) form a class (proper or no).

Then if we deal with metacategory $\mathbf A$ we say that Ob$(\mathbf A)$ is a conglomerate and Mor$(\mathbf A)$ so is.

To form a conglomerate we require:

every class is a conglomerate,
for every “property” P, one can form the conglomerate of all classes with property P,
conglomerates are closed under analogues of the usual set-theoretic constructions, and
the Axiom of Choice for Conglomerates; namely for each surjection between conglomerates $f:X\rightarrow Y$, there is an injection $g:Y\rightarrow X$ with $f\circ g=id_Y$.

So the question is: can we continue this hierarchy further to form the metacategory of all metacategories and so on?

This use of the term "quasicategory" is horribly archaic; please avoid it. — Mike Shulman, Oct 05 '17 at 15:48
Mike is too polite to self-advertise, but he wrote a great exposition on (set-theoretical) foundational issues in category theory: https://arxiv.org/abs/0810.1279 — Myself, Oct 05 '17 at 16:05
You should also be careful with the adjective "regular" here. Perhaps "usual" would be more appropriate. It's not a big deal, but I was a bit puzzled at first. — Arnaud D., Oct 06 '17 at 12:40
Metacategory is perhaps a suitable alternative to the ambiguous term "quasicategory". — David Roberts, Oct 06 '17 at 22:03
@DavidRoberts, now I checked the references and saw that term quasicategory is now being used in different meaning. So yes, metacategory is more suitable. — A. G, Oct 06 '17 at 22:51
@A. I have edited to refer to metacategories, and also fixed the list markdown syntax — David Roberts, Oct 07 '17 at 05:06
If one takes "can we continue" in this OP to be roughly the same as "do we still have well-defined operations at our disposal", then this MO thread definetely is very relevant (and was heretofore unlinked). A recommendation in an answer in that thread is roughly the same as 'Myself''s, i.e., to read M. Shulman: Set theory for category theory. arXiv:0810.1279. — Peter Heinig, Oct 07 '17 at 07:11
@PeterHeinig, yes, I’ve talked about the operations which I mentioned in my question, so you are right. — A. G, Oct 07 '17 at 09:16

The Hierarchy of “collections” for Categories

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