For already some time I am slightly bothered by the following question about endofunctors of the category of sets: Is there an endofunctor of set which maps each infinite cardinal $\kappa$ to a set of size $\kappa^+$ (i.e., the first larger cardinality)?
It is clear that under GCH, an example of such a functor is the power set functor as $2^\kappa = \kappa^+$. But without this axiom, I don't know of any other example. Anyway, I was interested whether one can construct such functor without any additional set axioms. To rephrase the question once mo: Is it always true under ZFC that there exists an endofunctor of the category of sets that maps $\kappa$ to a set of size $\kappa^+$?
My master thesis advisor suggested that such functor might be found as a subfunctor of the powerset functor, but I never managed to make it work. Am I missing something?