I was thinking about how to prove that certain classes of "sets with structure" (Groups, Rings, Topological spaces ...) are proper classes and not sets.
Once one notices that the collection of all finite sets is a proper class, one has a machinery to prove that any concrete category where finite sets are automatically objects will have a proper class of objects.
This for instance holds in the case of Metric Spaces and Topological Spaces, but what can we say about this problem in general?