In Bonn, we've been have a discussion on the topic in the title:

Suppose that A and B is are classes and that there are injections from A to B and fom B to A. Does it follow that there is a bijection between A and B?

Example: Let A the class of sets of cardinality one and let B be the class of sets of cardinality two. There is an injection

A -> B sending a to {a, emptyset},

B-> A sending b to {{b}}.

Does it follow that there is a bijection between A and B?