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

Suppose that $A$ and $B$ are classes and that there are injections from $A$ to $B$ and from $B$ 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

$A\to B$ sending $a$ to $\{a,\varnothing\}$,

$B\to A$ sending b to $\{\{b\}\}$.

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