Recently I posted this question https://mathoverflow.net/q/277582/41291 to mathoverflow and now it occurred to me that most likely I can get a quick answer here. Are there geometric morphisms f: YY -> XX which are (1) locally but not globally bounded, or (2) locally but not globally presheaf, or (3) as in (2) and bounded? In more detail, I mean this: there must be an object X in XX with global support (X->1 epic) such that the pullback f/X: YY/f^*(X) -> XX/X is (1) bounded, while f is not bounded, or (2) equivalent over XX/X to the topos (XX/X)^{CC^op} of internal presheaves on some internal category CC of XX/X, while YY is not equivalent to any such over XX, or (3) same as (2) and in addition f bounded. Can any of these happen? Hoping, Mamuka [For admin and other information see: http://www.mta.ca/~cat-dist/ ]