What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. preserves finite limits and power objects—hence also finite colimits and exponentials)?
On the lower end, Andreas Blass proved ("Exact functors and measurable cardinals") that there exists a nontrivial exact endofunctor of Set (that is, preserving finite limits and colimits) iff there exists a measurable cardinal. Since logical functors are a fortiori exact, the existence of a measurable cardinal is a necessary condition. On the upper end, any nontrivial elementary embedding j:V→V surely induces a logical endofunctor of Set, so the existence of a Reinhardt cardinal is a sufficient condition. But can it be pinned down more precisely?