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?