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](https://mathoverflow.net/questions/195178/an-example-of-two-cofibrant-dg-categories-whose-tensor-product-is-not-cofibrant)$^1$, 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)?
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**Footnotes.**
1. 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.
2. 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.
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This question is very much related to my question 2 [here](https://mathoverflow.net/questions/384362/relationship-between-different-definitions-of-the-hochschild-homology). 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.
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[1] Toën, B. [The homotopy theory of dg-categories and derived Morita theory.](https://arxiv.org/abs/math/0408337)
[2] Tabuada, G. [Théorie homotopique des DG categories.](https://arxiv.org/abs/0710.4303)
[3] Cohn, L. [Differential Graded Categories are k-linear Stable Infinity Categories.](https://arxiv.org/abs/1308.2587)
[4] Lurie, J. [Higher Algebra.](https://www.math.ias.edu/~lurie/) (September 18, 2017 version).