For any geometric morphism $f:\mathcal{F} \to \mathcal{E}$ of Grothendieck 1-topoi, there exists a functor of small categories $\ell :D\to C$ and left exact localizations $\mathcal{F} \hookrightarrow \mathcal{P}D$ and $\mathcal{E} \hookrightarrow \mathcal{P}C$ such that the inverse image functor $f^\* : \mathcal{E} \to \mathcal{F}$ is the composite


$$ \mathcal{E} \hookrightarrow \mathcal{P} C \xrightarrow{\ell^\*} \mathcal{P}D \to \mathcal{F}.$$


Is this also true for $\infty$-topoi?


The proof of the 1-topos fact that I know doesn't seem to generalize obviously. It's in section C2.5 of *Sketches of an Elephant*: we represent $\mathcal{F}$ as internal sheaves on an internal site in $\mathcal{E}$, then apply a Grothendieck construction to externalize that site relative to some site for $\mathcal{E}$, put a topology on it, and show that the resulting fibration is cover-reflecting and induces the given geometric morphism. The use of Grothendieck topologies is what it's not immediately clear how to generalize, since not every $\infty$-topos is a topos of sheaves in the straightforward sense.



Fibrations of sites for $\infty$-topoi
Here is an argument for the 1-categorical version that essentially bypass the use of internal site and should be much easier to generalize to the $\infty$-categorical case. ( I mean you can still see internal site barely hidden in plain sight, but the point is you don't need to see them to follow the proof)


As a first approximation, let's cheat a little, and allow ourselves to work with big sites - and we will deal with size issues at the end.


So how would we do this? A naive answer is to take $C = \mathcal{E}$ and $D = \mathcal{F}$... but that actually doesn't work at all.


The right way to do it is inspired from this idea of internal sites: One takes $C = \mathcal{E}$, but we take $D$ to be the comma category whose objects are triplets $X \in \mathcal{E}$, $Y \in \mathcal{F}$ and an arrow $u: Y \to f^\*X$.


(note that it corresponds to taking the internal site $\mathcal{F}$ seen as an $\mathcal{E}$-indexed category and applying the Grothendieck construction to it, exactly as in the argument you explained)


$l$ is the obvious forgetful functor $D$ to $C$, and the important point is that it has fully faithful right adjoint $i$ sending $X$ in $C = \mathcal{E}$ to the triplet $f^\* X \to f^\* X$.


Note that presheaves on $\mathcal{F}$ can be seen as special presheaves on $D$, the one that only depends on the $Y$ component, and (ignoring the obvious size problems) the left adjoint to this inclusion can be seen to be left exact (**edit:** this is because it can be identified with $f\_!$ for $f:D \to \mathcal{F}$ the forgetful functor which is itself left exact). In particular, "sheaves" over $\mathcal{F}$ are a left exact localization of presheaves over $D$.


Now, There is an obvious commutative square with vertically $i^\*$ and $f\_\*$ and horizontally the inclusion of $\mathcal{E}$ and $\mathcal{F}$ to $\mathcal{P}C$ and $\mathcal{P}D$.


The key point now is because $i$ is right adjoint to $l$, then $i^\*$ is right adjoint to $l^\*$, that is $i^\* = l\_\*$.


Taking the left adjoint of all the functors in the square above (and ignoring the size problem it creates at this stage) you get a commutative square with the two sheafification functors $\mathcal{P} C \to \mathcal{E}$ and $\mathcal{P} D \to \mathcal{F}$ horizontally and $l^\*$ and $f^\*$ vertically.


Restricting this to an object of $\mathcal{P} C$ that is already a sheaf, you get the desired decomposition.


Finally, let's deal with the size problems: as usual in this kind of argument, one can simply take $\kappa$ a regular cardinal such that:


* both $\mathcal{E}$ and $\mathcal{F}$ are locally $\kappa$-presentable.
* The category $\mathcal{E}\_\kappa$ and $\mathcal{F}\_\kappa$ of $\kappa$-presentable objects are closed under finite limits.
* $f^\*$ sends $\kappa$-presentable objects to $\kappa$-presentable objects.


Then taking $C = \mathcal{E}\_\kappa$ and $D$ the comma category for $f^\*: \mathcal{E}\_\kappa \to \mathcal{F}\_\kappa$ and you can make all the arguments above.


Obviously there are some difficulties to make all this works in $\infty$-category theory and I feel it would be too much works to to everything here. But I don't see any major obstacle, let me know if I missed one, I'd happy to look into it!


