In other words, is a bijection with a proper class a sufficient condition for a class to be a proper class?
The question just arose as I was learning about ordinals and cardinals and the associated paradoxes (Cantor's, Burali-Forti) that come when one tries to do something like talk about the size of the collection of all cardinals.
It seems that the motivation for calling these collections proper classes is that because when treated like a set they entail paradoxes, but the paradoxes seem to be specific to the nature of the set. The paradoxes don't seem to arise from the fact that they are "too large" in some sense to be called a set. Thus the question arose as to whether being "large enough" is a sufficient condition for a collection to be a proper class. In other words, is being a proper class about size or something more than size?