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?

ISin HTT; it's an etale geometric morphism. The fact I claimed is Proposition 2.2.1 here: arxiv.org/abs/1312.2204 $\endgroup$