Question
Let $\mathcal{C}$ be a small symmetric monoidal category. The category $\mathsf{sSet}^{\mathcal{C}}$ of simplicial precosheaves on $\mathcal{C}$ is a symmetric monoidal model category with respect to the projective model structure and the Day convolution product $\otimes$ (see [1, Theorem 4.1]). I want to know if cofibrant objects in $\mathsf{sSet}^{\mathcal{C}}$ are flat, i.e., tensor products by cofibrant objects preserve weak equivalences. Does anyone know a proof or a reference of this?
Remarks
Remark. By the standard argument (in terms of cell attachments), it will suffice to show that each functor of the form $h_c\otimes-$ preserves weak equivalences, where $h_\mathcal{c}$ denotes the precosheaf corepresented by $c$.
Remark. A stronger statement appears in, [1, Theorem 4.1 (b)] where the authors assert that weak equivalences of $\mathsf{sSet}^{\mathcal{C}}$ is stable under Day convolution. However, the argument relies on the following auxiliary assertion, which I do not know how to prove:
($\ast$) Let $i_!:\prod_{c\in\mathcal{C}}\mathsf{sSet}\rightleftarrows\mathsf{sSet}^\mathcal{C}:i^*$ denote the left Kan extension-restriction adjunction. For every pair of objects $X,Y\in \mathsf{sSet}^{\mathcal{C}}$, the object $i^*(X\otimes Y)$ is a retract of $i^*((i_!i^*X)\otimes (i_!i^*Y))$ (and the retraction is natural in $X$).
In fact, condition ($\ast$) implies that, for each $c\in \mathcal{C}$, there is a map
$$ \operatorname{colim}_{{c_1\otimes c_2\to c}}X(c_1)\times Y(c_2)\to \coprod_{a,b\in \mathcal{C}}\mathcal{C}(a\otimes b,c)\times X(a)\times Y(b). $$
But I'm not seeing any natural such map. I would appreciate it if someone can clarify ($\ast$), too.
Update (2024.11.30) After a few correspondences with Michael Batanin, it became clear that (*) is an oversight. So the proof of [1, strong h-monoidality part of Theorem 4.1(b)] does not work (although one can prove the plain h-monoidality part of the theorem by a different argument).
Motivation (if anyone is interested)
In a recent work, Bayındır and Chorny generalized Dugger's work on universal homotopy theories and showed that every simplicial, combinatorial symmetric monoidal model category is equivalent (in an appropriate sense) to a monoidal left Bousfield localization of $\mathsf{sSet}^{\mathcal{C}}$ for some $\mathcal{C}$. [2]
For some reason or other, I want to show that the localization satisfies the monoid axiom. There is a fairly general theorem, due to White, which gives a sufficient condition for localizations to inherit the monoid axiom [3, Theorem 8.9]. However, this theorem requires cofibrant objects to be flat prior to localizing, hence this question.
References
[1] Batanin, M., & Berger, C. (2017). Homotopy theory for algebras over polynomial monads. Theory Appl. Categ., 32, Paper No. 6, 148–253, http://www.tac.mta.ca/tac/volumes/32/6/32-06.pdf.
[2] Bayındır, H., & Chorny, B. (2023). Admissible replacements for simplicial monoidal model categories. Algebraic & Geometric Topology, 23(1), 43–73, https://doi.org/10.2140/agt.2023.23.43.
[3] White, D. (2022). Monoidal Bousfield localizations and algebras over operads. In Equivariant topology and derived algebra (Vol. 474, pp. 180–240). Cambridge Univ. Press, Cambridge, https://arxiv.org/abs/1404.5197.
