$\newcommand{\M}{\mathcal{M}}$

Suppose I have a monoidal simplicial model category in which every object is cofibrant $(\M,\otimes,\mathbb{1})$ and I want to look at its underlying monoidal quasicategory, which I'll write as $N(\M)$, the simplicial nerve of $\M$. One way to do this is the following construction (following Variant 4.1.3.17 of Lurie's Higher Algebra):

First take the sub-simplicial monoidal category of fibrant and cofibrant objects of $\M$, denoted $\M^\circ$. This is still a monoidal simplicial category. Now produce its so-called "category of operators," i.e. the free semi-cartesian monoidal category on its underlying simplicial multicategory. To be explicit, do the following:

Let $Multi(\M^\circ)$ be the simplicial multicategory whose objects are the objects of $M^\circ$ and whose "multimapping" objects are $$Mul(\{m_1,\ldots,m_k\},m)=\coprod_{\alpha\in\Sigma_k} Hom_{\M^\circ}(m_{\alpha(1)}\otimes\cdots\otimes m_{\alpha(k)},m).$$ Now produce the category of operators of this multicategory, which I'll write as $(\M^\circ)^\otimes$. This category has as objects pairs $(\langle k\rangle,\{m_1,\ldots,m_k\})$ where $\langle n\rangle$ is the finite pointed set $\{\ast,1,\ldots,k\}$ and $\{m_1,\ldots,m_k\}$ is a finite list of objects of $Multi(\M^\circ)$ (hence a finite list of objects of $\M^\circ$). This category has mapping objects given by $$Hom_{(\M^\circ)^\otimes}((\langle k\rangle,\{m_1,\ldots,m_k\}),(\langle j\rangle\{l_1,\ldots,l_j\}))=\coprod_{f:\langle k\rangle\to\langle j\rangle}\prod_{1\leq r\leq j}Mul(\{m_i\}_{i\in f^{-1}(r)},l_r).$$

This construction probably seems a bit unwieldy, but the point is that this final category $(\M^\circ)^\otimes$ admits a forgetful functor $(\M^\circ)^\otimes\to Ass^\otimes$, the category of operators of the associative operad. In this case, since all of our mapping objects are Kan complexes (this follows from the fact that we took bifibrant objects and that cofibrant objects are closed under tensor product), we get that $N((M^\circ)^\otimes)\to N(Ass^\otimes)$ defines a monoidal structure on the quasicategory $N(M^\circ)$.

Now, my question is about the following: Suppose my simplicial monoidal model category $\M$ has a tensor product that preserves *fibrant* objects in addition to cofibrant objects. Then take the opposite category of $\M^\circ$, which we'll write $(\M^\circ)^{op}$, which is still a simplicial monoidal category. Moreover, since the tensor product preserves fibrancy, the mapping objects of $((\M^\circ)^{op})^\otimes$ are Kan complexes and so we still have a structure map of quasicategories $N(((\M^\circ)^{op})^\otimes)\to N(Ass^\otimes)$ defining a monoidal structure on $N((\M^\circ)^{op})$.

My question is the following: is the monoidal structure so defined on $N((\M^\circ)^{op})$ equivalent to the monoidal structure on $N(M^\circ)^{op}$ obtained by using the Grothendieck construction to get a functor $N(Ass^\otimes)\to Cat_\infty$, composing with $op$ to get a functor $N(Ass^\otimes)\to Cat_\infty\overset{op}\to Cat_\infty$ and then reversing the Grothendieck construction?

----(EDIT)----

Perhaps a bit of background. The main point of this whole thing here is to take a simplicial monoidal model category and lift strict comonoids and comodules therein to its associated monoidal quasicategory (in the sense of Lurie). These can be relatively easily defined to be strict monoids and modules in the opposite category, but one runs into difficulties using Lurie's set-up to access the monoidal structure on the nerve of the opposite category. When the tensor product preserves fibrancy it seems like the above construction might be a work around. At the very least, it lets you define comodules and comonoids in SOME monoidal quasicategory, but it's not clear that this is equivalent to the canonically defined (up to contractible space of choices) opposite monoidal quasicategory. If anyone has some other idea for lifting comonoids and comodules, I'd be very interested to know about it.

beforetaking nerves of anything: there should still be a (monoidal 2-)functor $Ass^\otimes \to Cat_\Delta$ expressing the monoidal structure on $\mathcal{M}$, and the composite with the "opposite" construction classifies the monoidal structure on $\mathcal{M}^{op}$. $\endgroup$ – Tyler Lawson Mar 27 '17 at 0:21