TITLE: Sheaves over a sheaf
QUESTION [6 upvotes]: Everything I write I mean in the in the sense of Lurie's HTT. 
Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C})_{/F}$ is equivalent to $ Sh( \mathcal{C}_{/F}) \subseteq Fun(\mathcal{C}^{op}_{/F} , \mathcal{S})$? (The case I'm interested in is when $\mathcal{C}$ is $Aff$ or $derAff$) If this is discussed in HTT or elsewhere I'd be super happy with a reference. Thanks!

REPLY [5 votes]: The version for presheaves is proven in 
HTT, 5.1.6.12
To show that this equivalence carries over to sheaves, you need to think about how to pull the Grothendieck topology back to the category of elements.  I don't have a precise reference yet, but it should be somewhere around there. I'll take a look in the morning.
Somehow, Grothendieck topologies in Lurie's setup only rely on data in the homotopy category, so it shouldn't be surprising that this should now work by transfer of results from classical sheaf theory, but again, I have to find a precise reference for you.