Let $k$ be a commutative ring with $1$, and let $\mathsf{dgCat}_k$ be the category of $k$-linear dg-categories, as defined in [1, Section 2]. We may equip $\mathsf{dgCat}_k$ with the Morita model structure [2, Théorème 2.27], which we will denote by $\mathsf{dgCat}_k^\textrm{Mor}$.
The category $\mathsf{dgCat}_k$ is symmetric monoidal: if $\mathcal{C},\mathcal{D}\in\operatorname{Obj}(\mathsf{dgCat}_k)$, then we define $\mathcal{C}\otimes_{\mathsf{dgCat}_k}\mathcal{D}$ to be the $k$-linear dg-category with objects $$ \operatorname{Obj}(\mathcal{C}\otimes_{\mathsf{dgCat}_k}\mathcal{D}) := \operatorname{Obj}(\mathcal{C})\times\operatorname{Obj}(\mathcal{D}) $$ and morphism complexes $$ \left(\mathcal{C}\otimes_{\mathsf{dgCat}_k}\mathcal{D}\right)((X,Y),(X',Y'))_* := \mathcal{C}(X,X')_*\otimes_k \mathcal{D}(Y,Y')_*. $$
While the tensor product defined above does not respect the Morita model category structure$^1$, we may derive it to obtain a symmetric monoidal model category structure on $\mathsf{dgCat}_k^{\textrm{Mor}}$, which we will denote by $\left(\mathsf{dgCat}_k^\textrm{Mor},\otimes_{\mathsf{dgCat}_k^\textrm{Mor}}^L\right)$ (see [2, Remarque 2.40]). Explicitly, $$ \mathcal{C}\otimes_{\mathsf{dgCat}_k^\textrm{Mor}}^L\mathcal{D} \simeq Q(\mathcal{C})\otimes_{\mathsf{dgCat}_k}\mathcal{D}, $$ where $Q$ is a cofibrant replacement functor for the $\mathsf{dgCat}_k^{\textrm{Mor}}$.
Cohn has shown [3, Corollary 5.5] that the underlying $\infty$-category as defined in [4, Definition 1.3.4.15]$^2$ of $\mathsf{dgCat}_k^\textrm{Mor}$ is equivalent to the $\infty$-category of small idempotent-complete $k$-linear stable $\infty$-categories: $$ N(\mathsf{dgCat}_k^\textrm{Mor})[W^{-1}]\simeq\operatorname{Mod}_{\operatorname{Perf}(Hk)}((\mathcal{Cat}_\infty^\textrm{perf})^\otimes). $$ In fact [3, Corollary 5.7], it is also equivalent to the $\infty$-category of compactly-generated presentable $k$-linear stable $\infty$-categories with functors that preserve colimits and compact objects: $$ N(\mathsf{dgCat}_k^\textrm{Mor})[W^{-1}]\simeq\operatorname{Mod}_{Hk\textrm{-}\operatorname{Mod}}((\mathcal{Pr}_{\textrm{st},\omega}^L)^\otimes). $$
We also have symmetric monoidal structures on both these $\infty$-categories. The Lurie tensor product [4, Proposition 4.8.1.15] induces a symmetric monoidal structure on $\operatorname{Mod}_{\operatorname{Perf}(Hk)}((\mathcal{Cat}_\infty^\textrm{perf})^\otimes)$ and $\operatorname{Mod}_{Hk\textrm{-}\operatorname{Mod}}((\mathcal{Pr}_{\textrm{st},\omega}^L)^\otimes)$ (I believe this follows from [4, Theorem 3.3.3.9], at least in the case of $\operatorname{Mod}_{Hk\textrm{-}\operatorname{Mod}}((\mathcal{Pr}_{\textrm{st},\omega}^L)^\otimes)$), and the symmetric monoidal structure $\otimes^L_{\mathsf{dgCat}_k^\textrm{Mor}}$ induces a symmetric monoidal structure on $N(\mathsf{dgCat}_k^\textrm{Mor})[W^{-1}]$ (see [4, Example 4.1.7.6]).
My question: is the equivalence above an equivalence of symmetric monoidal $\infty$-categories? In other words, does the symmetric monoidal model category structure on $\mathsf{dgCat}_k^\textrm{Mor}$ induce the expected symmetric monoidal structure given by the Lurie tensor product on the underlying $\infty$-category of dg-categories? If the symmetric monoidal structures do not coincide, what is the relationship between them (if any)?
Footnotes.
- The linked MO question deals with the Dwyer-Kan model structure on $\mathsf{dgCat}_k$ (see [1, Section 2] and [2, Théorème 1.8]), but the issue is present in the Morita model structure as well.
- It appears Cohn does not restrict to cofibrant objects as in the cited definition, but [4, Remark 1.3.4.16] implies that this is not a problem, as the Morita model structure on $\mathsf{dgCat}_k$ admits functorial fibrant-cofibrant factorizations of morphisms.
This question is very much related to my question 2 here. In some sense, both are about the same underlying question/confusion, but I wanted to ask a more focused question about the symmetric monoidal structures in particular. I also hope that the (perhaps excessive) citations will be useful to anyone attempting to make their way through this literature and find precise definitions and references for the first time.
[1] Toën, B. The homotopy theory of dg-categories and derived Morita theory.
[2] Tabuada, G. Théorie homotopique des DG categories.
[3] Cohn, L. Differential Graded Categories are k-linear Stable Infinity Categories.
[4] Lurie, J. Higher Algebra. (September 18, 2017 version).
