When the global section functor is a Cartesian fibration?

Question

Given a Cartesian fibration $p : \mathbf{E} \to \mathbf{B}$ over an $\infty$-topos the paper by Marc Hoyois mentioned in his answer to this question gives some sufficient conditions for $\mathbf{E}$ to be an $\infty$-topos. I'd like to know when the converse holds. That is, if we have a functor between toposes $p : \mathbf{E} \to \mathbf{B}$, when it is a Cartesian fibration? Since this question is probably too general, it may be assumed that $p$ is the direct image of a geometric morphism and even more that it is the global section functor $\mathrm{Hom}(1,-) : \mathbf{E} \to \mathrm{\infty Grpd}$.

More generally, if $\mathbf{C}$ is an arbitrary $\infty$-category with a terminal object, when the functor $\mathrm{Hom}(1,-) : \mathbf{C} \to \mathrm{\infty Grpd}$ is a Cartesian fibration? Since this question is also probably too general, it may be assumed, if it helps, that $\mathbf{C}$ is locally presentable and that $\mathrm{Hom}(1,-)$ has a right adjoint.

Answer 1

In fact I believe something quite general can be said: if $p:C\to D$ is any functor that preserves finite limits, then it is a Street fibration if and only if it has a fully faithful right adjoint. A Street fibration is just a functor whose isofibrant replacement is a cartesian fibration in the usual sense; this is the only sensible thing to ask about a functor between given toposes, since toposes are really only well-defined up to equivalence of categories. A sketch of the proof of this is here, and I think it should generalize straightforwardly to $(\infty,1)$-categories.

If $p$ is the direct image functor of a geometric morphism, then it has a fully faithful right adjoint if and only if that geometric morphism is local, a familiar topos-theoretic property.