Definition of Category

In Spanier's book of algebraic topology, there is a definition of "categories" which entails "a class of objects".

I realize that the vagueness of the concept of "class of objects" is exactly used instead of "set of sets" because we want to avoid certain paradoxes of set theory.

Still, I am wondering, is there a more formal definition, or axiomatization, of what a "class of objects" mean in the concept of a category?

The notion of a class is defined rigorously in Von Neumann–Bernays–Gödel set theory, which is a conservative extension of ZFC.

Basically, you can form classes of sets using unrestricted comprehension, and you can freely take subclasses, images of classes under functions, and use the Axiom of Choice on classes of sets. However, no proper class is allowed to be an element of anything -- if a class $C$ is an element of a class $D$, then $C$ must be a set. In particular, there is no class of all classes, although there is a class of all sets.

It is just that, a class. That is: Given $A$, it is either true or false that $A\in\operatorname{Obj}(\mathcal C)$. Put differently, we may assume that "is an object of category $\mathcal C$" is a valid predicate.