Unaugmentable cosimplicial simplicial sheaves and realization functor

I'm studying the construction of the $\mathrm{Sing}$ functor in Morel-Voevodsky $\mathbb{A}^1$-homotopy theory of schemes'' and I was trying to understand the properties of its left adjoint, the realization functor.

To start with, let $\mathcal{C}$ be a site (with an interval, in the sense of M-V) and let $\mathsf{Sh}(\mathcal{C})$ (resp. $\mathsf{sSh}(\mathcal{C})$) be the category of sheaves (resp. simplicial sheaves) on $\mathcal{C}$.

The category of simplicial sheaves is enriched, tensored and cotensored over the category of sheaves in an obvious way. In particular, for a sheaf $F$ and a simplicial sheaf $X$ we can define the tensor product $F\otimes X$ of $F$ and $X$ as the product of the constant simplicial sheaf at $F$ with $X$.

Now, if $X$ is a simplicial sheaf and $D^\bullet$ is a cosimplicial simplicial sheaf, we define the tensor product $X\otimes_{\Delta^{\text{op}}} D^\bullet$ of $X$ and $D^\bullet$ (or realization of $X$ with respect to $D^\bullet$, denoted $|X|_{D^\bullet}$ in Morel-Voevodsky) as the coend of the functor:

\begin{align*} X\otimes D^\bullet \colon \Delta^{\text{op}}\times\Delta&\to \mathsf{sSh}(\mathcal{C})\\ ([n],[m])&\mapsto X_n\otimes D^m \end{align*}

This construction gives rise to a bifunctor:

$$\otimes_{\Delta^{\text{op}}}\colon \mathsf{sSh}(\mathcal{C})\times \mathsf{sSh}(\mathcal{C})^{\Delta}\to \mathsf{sSh}(\mathcal{C})$$

Now, a cosimplicial simplicial sheaf $D^\bullet$ is said to be unaugmentable if the morphism $(d^0,d^1)\colon D^0\amalg D^0\to D^1$ is a monomorphism. Equivalently, $D^\bullet$ is unaugmentable iff the equalizer of $d^0$ and $d^1$ is empty. Those are the cofibrant objects in the Reedy model structure on $\mathsf{sSh}(\mathcal{C})$, for any injective model structure on simplicial sheave.

Morel and Voevodsky prove that a cosimplicial simplicial sheaf $D^\bullet$ is unaugmentable if and only if the functor $-\otimes_{\Delta^{\text{op}}}D^\bullet$ preserves monomorphisms. Moreover, for some specific unaugmentable $D^\bullet$ they show that the functor $-\otimes_{\Delta^{\text{op}}}D^\bullet$ also preserves $I$-local equivalences (where $I$ is an interval in the sense of M-V and we localize the local-injective model structure on simplicial sheaves with respect to the morphism $I\to *$).

Question: Is the functor $\otimes_{\Delta^\text{op}}$ defined above a left Quillen bifunctor? Here, we put the $I$-local model structure on the category of simplicial sheaves (in particular cofibrations are monomorphisms) and we put the Reedy model structure on the category of cosimplicial objects in the $I$-local model category of simplicial sheaves (In particular cofibrant objects are unaugmentable cosimplicial simplicial sheaves).

• An interesting question, will get back to it. For the time being, let $I'$ be another interval, and let $D_{I'}$ be the cosimplicial simplicial sheaf built from $I'$. If $\otimes_\mathrm{\Delta^{\!op}}$ is a left Quillen bifunctor, then the functor $\otimes_\mathrm{\Delta^{\!op}}D_{I'}$ preserves monomorphisms that are $I$-weak equivalences. Lemma 3.12 in M-V does not apply to $\otimes_\mathrm{\Delta^{\!op}}D_{I'}$, so you may look for counterexamples instead. – user337830 May 25 '17 at 15:01
• In particular, take $I'$ to be a space that not $I$-contractible, like an elliptic curve, or simply the disjoint union of two copies of the terminal space, and apply the functor $\otimes_\mathrm{\Delta^{\!op}}D_{I'}$ to monomorphisms that are $I$-weak equivalences between representable simplicial sheaves (where it is easy to calculate). – user337830 May 25 '17 at 15:01
• @user337830 Thank you for the comment, very enlightening indeed! – Stefano Nicotra May 26 '17 at 13:56