Doesn't answer the question posed on categories but just has a jab at the fact that some categories are much too large to fit in the usual ZFC meta-theory. It's why most books on category theory will mention universes or classes and then just forget about the issue entirely. Type theory has the same problem, to talk about the type of all types requires introducing a hierarchy of universes to avoid logical inconsistencies, i.e. if U denotes the type of all types then U:U would make the theory inconsistent which is the same as saying the set of all sets is a set. Ultimately, to avoid the issue of circular self-reference you must have some kind of hierarchy of classes or universes or you might accidentally introduce a circular self-reference when quantifying over all sets, types, objects, etc.
The issue is the notion that set theory can be the ultimate foundation of mathematics. ZF set theory is not a good foundation because it says “too much” about sets. For examples, every set in ZF has to have a power set, which means you can never talk about any object that doesn’t have a power set, such as the class of all Rings, or the surreal numbers. As it turns out, quite a lot of objects in mathematics are too large to be sets.
Therefore the ultimate foundation must be something else, such as categories, classes and types. You can still have ZF set theory but not at the very bottom.
