CAUTION: The Sender of this email is not from within Dalhousie. As is well known a geometric morphisms F --| U : EE -> SS is locally connected iff F preserves dependent products. But has anyone come across a g.m. whose inverse image part preserves ordinary function spaces, i.e. exponentials, but not dependent function spaces? In connection with Lawvere and Menni's TAC 30 paper (section 10) this question has been asked under the additional assumption that the g.m. is also hyperconnected and local. Triggered by discussion with Matias Menni I have thought about it with moderate success and then found out that I even do not know the answer to the (possibly) simpler question formulated at the beginning of this message. My hope now is that someone happens to know a counterexample to this simpler question at least... In my eyes the requirement that the further left adjoint be fibered/indexed is most natural if one thinks of gm's as cocomplete and locally small toposes over an arbitrary base topos (which I do as is apparent from my notes on Fibered Categories (section 18)). All examples of precohesive gm's I know of are locally connected. But it would be very surprising if this were always the case (unless the base is Set where essential and locally connected are known to coincide). Thanks in advance, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]