(By the above, I mean to ask, what fibred category do they intend, when they write $U\to\mathfrak{X}$)

@Απών Presumably they mean $\mathcal{C}_{/U}$, identified with $U$ via the Yoneda embedding from $\mathcal{C}$ to categories fibered in groupoids (in fact sets) over $\mathcal{C}$

I agree that the notation is a bit confusing in this context, although suppressing the Yoneda embedding is extremely common

Why are the inverse image part of geometric morphisms required to preserve final objects? It is completely reasonable that they preserve pull-backs; they already preserve colimits, and these are the two pieces of structure needed to talk about descent (in the sense of Rezk). As any functor which preserves pull-backs and final objects preserves finite limits, I wonder what desiderata in topos theory stop holding if we drop the requirement that inverse image preserve final objects.

I should maybe add, that I have definitely already proved stuff using this property... it's just that this property seemed mostly convenient rather than interacting in some canonical way with fundamental properties of toposes.

@AdrianClough you are essentially asking why X_{/U}-->X is not a geometric morphism right? I think these morphisms certainly make sense and also very much considered in HTT, so I guess it is more of a notational convension. I mean it seem to make perfect sense to talk about the category of topoi with morphisms generated by geometric and etale ones i think.

@AdrianClough This might be a cop out, but without that condition the embedding of (sober) topological spaces is not fully faithful anymore...

Hey y'all, so in Lemma 5.2.2.29 of HA, Lurie says that Mod_A^{Ass}(C), for A an associative algebra, and C a monoidal ∞-category, is a monoidal category. But we can't generally assume this unless C has the property that its tensor product preserves geometric realizations right?

Someone asked me this question and I can't see any way to fix it other than to assume Lurie just forgot to write in that assumption. I don't know if this affects anything else down the line.