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Morgan Rogers and I were able to construct a counterexample of the form PSh(M) --> PSh(N) where M and N are monoids. It arises from our joint work-in-progress in which we study exactly these kind of geometric morphisms, in a systematic way. After talking about it with Thomas Streicher, we have now written up the counterexample in more detail. You can find it here: https://arxiv.org/abs/2009.12241 (3 pages).

Dear Jens and Morgan, thanks a lot for your very nice counterexample which I never would have found on my own! Alas, it leaves open Lawvere and Menni's question whether precohesive geometric morphisms are always also locally connected. Does your toolbox also provide a counterexample to this implication. The current one does not do this job since you show the leftmost adjoint does not preserve binary products. (By Lemma 2.7 of Johnstone's 2011 TAC paper preservation of binary products by the leftmost adjoint is equivalent to preservation of exponentials by the inverse image part for hyperconnected and local geometric morphisms.) BTW I think that locally connected, hyperconnected and local is the correct generalization of essential, 2-valued and local from base Set to arbitrary base toposes from the point of view of fibered categories. Best, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]