CAUTION: The Sender of this email is not from within Dalhousie. I should have added that the question I asked today is very related to the one raised by Lawvere and Menni at the end of section 10 of their TAC 30 paper from 2015. In my eyes the relation is as follows. Even if not all essential hyperconnected local gm's are locally connected there still remains the question whether the implication holds under the additional assumption that the inverse image part of the gm p : EE-->SS preserves (ordinary) exponentials. Of course, locally connected amounts to p^* preserving dependent function spaces (the Pi_u for u in SS. So the question is whether for the purpose at stake it suffices that p^* preserves ordinary (non-dependent) function spaces. Thomas S. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]