Let $\mathcal{V_1}$ and $\mathcal{V_2}$ be cocomplete symmetric monoidal categories, each endowed with a cosimplicial object $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. Denote by $|-|=|-|_{\mathcal{V}_i}:s\mathcal{V}_i\to \mathcal{V}_i$ the functor tensor product $-\otimes_\Delta \Delta^\bullet_{\mathcal{V}_i}$, i.e. the coend

$$|X_\bullet|=X_\bullet \otimes_\Delta \Delta^\bullet = \int^n X_n \otimes \Delta^n$$ for $X_\bullet\in s\mathcal{V}_i$ a simplicial object in $\mathcal{V}_i$.

Let $F:\mathcal{V}_1\to \mathcal{V}_2$ be a lax symmetric monoidal functor which is a left adjoint and such that $F(\Delta^\bullet_{\mathcal{V}_1})\cong \Delta^\bullet_{\mathcal{V}_2}$.

Then there is an induced natural transformation $$\tau:|F-|\Rightarrow F|-|$$ of functors $s\mathcal{V}_1\to \mathcal{V}_2$, since $F$ preserves coends and the chosen cosimplicial objects.

Now, the category $s\mathcal{V}_i$ has a symmetric monoidal structure given by the pointwise formula $(X \otimes Y)_n = X_n \otimes Y_n$, and it is such that the induced functor $F:s\mathcal{V}_1\to s\mathcal{V}_2$ is lax symmetric monoidal. Suppose further that $|-|:s\mathcal{V}_i\to \mathcal{V}_i$ is lax symmetric monoidal. Therefore $\tau$ is a natural transformation between lax symmetric monoidal functors, so it makes sense to ask:

Is $\tau$ a lax symmetric monoidal transformation? i.e., do the following diagrams commute?

I am under the impression that it won't be the case in general... Are there reasonable extra conditions to impose such that it will hold?

Here is an **example** of the above scenario, where I'm hoping the answer is affirmative (and it may thus serve as an inspiration to extract additional conditions to ensure a general $\tau$ above to be lax symmetric monoidal).

Consider $\mathcal{V}_1$ to be simplicial sets and $\mathcal{V}_2$ to be topological spaces. Endow the first one with the Yoneda embedding as a cosimplicial simplicial set, and endow the second one with the standard cosimplicial topological space. These objects yield internal geometric realizations: of a bisimplicial set into a simplicial set (which one can prove to be the diagonal functor), and of a simplicial space into a space (the standard one).

As a functor $F$, we will consider the standard "extrinsic" geometric realization of a simplicial set into a topological space, which satisfies all the hypotheses above. So the question in this case is whether the natural isomorphism $\tau_{X_{\bullet,\bullet}}: |\mathrm{diag} X_{\bullet,\bullet}|\cong |[n]\mapsto |X_{n,\bullet}||$ in $X_{\bullet, \bullet}\in ss\mathrm{Set}$ is (cartesian) symmetric monoidal.