In the definition of meta-categories, we have not imposed any restrictions on what kinds of things the objects and morphisms are; our definition is pre-mathematical, so we do not assume beforehand that there is a such thing as a collection of “all” meta-categories.
We may define analogous notions of meta-functor, etc. But we do not
assume that the notion of “all meta-functors \(\mathfrak {C}\to \mathfrak {D}\)” is well-defined; the notion is entirely schematic.
Assumption. We assume a meta-category \(\boldsymbol {\mathfrak {Coll}}\) whose objects we will refer to as “collections”. We assume that the meta-category of all collections satisfies the axioms of Lawvere’s ETCS.