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 $sPh(\mathcal{C})\_{proj}$.



Monoidal structure on simplicial sheaves
Yes. This is well-known and can be deduced, for example, from the general statements in the last section of [this paper](http://arxiv.org/pdf/0708.2067v2.pdf) of Barwick. Take $V$ to be the symmetric monoidal model category of simplicial sets. Note that it is [tractable](http://ncatlab.org/nlab/show/tractable+model+category) 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.


