Monoidal structure on simplicial sheaves

Question

Let $\mathcal{C}$ be a site and let $sPh(\mathcal{C})_{proj}$ be the category of simplicial presheaves equipped with the projective model structure. This category is a closed monoidal model category (with the ordinary tensor product of functor $-\times -$, and an internal hom $sPh(\mathcal{C})[-,-]$) and a simplicial model category.

Now let $sPh(\mathcal{C})_{proj, Cech}$ be the model structure obtained from the the global model structure on simplicial presheaves on $\mathcal{C}$ by left Bousfield localizations at Cech covers. Here my questions.

1) Is it again a simplicial model category? tensored and cotensored?

2) Is it again a closed monoidal model category?

3) In particular from nlab I know that for any cofibrant object $X$, the functor $X\times -$ is again a Quillen left in $sPh(\mathcal{C})_{proj, Cech}$ with right adjoint $sPh(\mathcal{C})[X,-]$. See http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#MonoidalStructure

Now let $\{D_{i}\}\to D$ be a cover of $D\in\mathcal{C}$ and let $C\{D_{i}\}$ its Cech nerve. Let $A$ be a fibrant object in $sPh(\mathcal{C})_{proj, Cech}$, then does the map between cofibrant objects

$$C\{D_{i}\}\to y(D), $$ where $y$ is the Yoneda embedding, induces a weak equivalence $$ sPh(\mathcal{C})[y(D),A]\to sPh(\mathcal{C})[C\{D_{i}\},A] $$ in $sPh(\mathcal{C})_{proj, Cech}$? I ask that because $C\{D_{i}\}\to y(D)$ is not in general a weak equivalence in $sPh(\mathcal{C})_{proj}$.

Answer 1

Yes. This is well-known and can be deduced, for example, from the general statements in the last section of this paper of Barwick. Take $V$ to be the symmetric monoidal model category of simplicial sets. Note that it is tractable because every object is cofibrant. By Corollary 3.33, the projective model structure on simplicial presheaves in $C$ is symmetric monoidal. Then by Theorem 3.36 the Cech-local projective model structure is simplicial, and by Theorem 3.38 it is symmetric monoidal.