For an -local site, the (∞,1)-sheaf (∞,1)-topos over it is a local (∞,1)-topos, in that the global section (∞,1)-geometric morphism has a further right adjoint (∞,1)-functor
We may present the (∞,1)-sheaf (∞,1)-topos by the local model structure on simplicial presheaves
For the notation see the details of the analagous proof at ∞-connected site. As discussed there, the functor is given by evaliation on the terminal object. At the level of simplicial presheaves the sSet-enriched right adjoint to is given by
as confirmed by the following end/coend calculus computation:
where in the second but last step we used the co-Yoneda lemma.
It is clear that
is a Quillen adjunction, since manifestly preserves fibrations and acyclic fibrations. Since is a left proper model category to see that this descends to a Quillen adjunction on the local model structure on simplicial presheaves it is sufficient to check that preserves fibrant objects, in that for a Kan complex we have that satisfies descent along Cech nerves of covering families.
This follows from the second defining condition on the -local site , that . Using this we have for fibrant the descent weak equivalence
where we use in the middle step that is a simplicial model category so that homming the weak equivalence between cofibrant objects into the fibrant object indeed yields a weak equivalence (using the factorization lemma).