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?