I have a very simple question regarding geometric morphisms of $\infty$ topoi, but have been unable to find the answer in Lurie's HTT (although it seems likely that its there somewhere and I just can't find it). 


Classically, given two sites $C$ and $D$, a morphism of sites $f:C\to D$ induces a geometric morphism of topos $f\_{\*}:Shv(C)\to Shv(D)$. Does the statement still hold in the $\infty$ context? 


What I'm really after is the following: Let $C$ be an $\infty$ site and let $c$ be an object of $C$. Then $C\_{/c}$ inherits a topology from $C$ by restricting the covering sieves to objects in the overcategory. The forgetful functor
$f:C\_{/c}\to C$ is trivially a morphism of $\infty$ sites. Does $f$ induce a functor
$$
f^\*:Shv(C)\to Shv(C\_{/c})$$
preserving finite $\infty$ limits?



Geometric morphism of $\infty$ topos
I never found any discussion about this in HTT either, but it turns out to work exactly the same way as in SGA 4.


That is, let $f^\* : P(D) \to P(C)$ denote the restriction functor on presheaves, and $f\_!$ its left adjoint (given by left Kan extension).
Since $f$ preserves covering families, $f\_!$ preserves local equivalences, so by adjunction $f^\*$ preserves sheaves and induces a functor
 $$ f^\*\_s : Sh(D) \to Sh(C). $$
Define
 $$ f\_!^s := a\_D f\_! i\_C : Sh(C) \to Sh(D), $$
where $i\_C$ is the inclusion $Sh(C) \hookrightarrow P(C)$ and $a\_D : P(D) \to Sh(D)$ is the associated sheaf functor.
It is straightforward to check that this is left adjoint to $f^\*\_s$.
The fact that $f\_!^s$ is left-exact follows from the fact that $i\_C$ commutes with (small) limits, $f\_!$ commutes with finite limits, and $a\_D$ commutes with finite limits (I am assuming of course that $D$ is small).


The fact that $f\_! : P(C) \to P(D)$ commutes with finite limits follows from the pointwise formula for the Kan extension
 $$ f\_! F(d) = \varinjlim\_{(c \in C, d \to f(c)) \in (d/f)} F(c) $$
where, since $f$ is left-exact, the comma category $(d/f)$ is filtered. (The commutativity of filtered colimits and finite limits in an $\infty$-topos is Example 7.3.4.7 in HTT.)


