Fibrations of sites for $\infty$-topoi 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.
 A: 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!
