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?