There are several notions of nerves, including nerves of categories, $2$-categories, and simplicial categories. These define functors \begin{align*} \mathrm{N} &\colon \mathrm{Cats}_{(2,1)} \to (\infty,1)\text{-}\mathrm{Cats},\\ \mathrm{N}^2 &\colon \mathrm{Bicats}_{(3,1)} \to (\infty,1)\text{-}\mathrm{Cats},\\ \mathrm{N}^s &\colon \mathrm{sCats}^{\text{loc. Kan}}_{(2,1)} \to (\infty,1)\text{-}\mathrm{Cats}. \end{align*} Taking the appropriate nerves for each of these functors, we obtain $(\infty,1)$-functors from the $(\infty,1)$-categories of
- Categories, functors, and natural isomorphisms;
- Bicategories, pseudofunctors, invertible pseudotransformations, and invertible modifications;
- Simplicial categories, simplicial functors, and simplicial natural isomorphisms;
to the $(\infty,1)$-category of $(\infty,1)$-categories.
All of the $(\infty,1)$-categories involved become symmetric monoidal $(\infty,1)$-categories when equipped with the Cartesian product. Can we endow the above $\infty$-functors with a symmetric monoidal structure as well?
Also, if we don't truncate to $(2,1)$-categories, we get $(\infty,2)$-functors from the $(\infty,2)$-categories of categories or simplicial categories to that of $(\infty,1)$-categories. Can we make these two $(\infty,2)$-functors into symmetric monoidal $(\infty,2)$-functors between symmetric monoidal $(\infty,2)$-categories as well?
(I won't ask the same for the second case above, since I gather it wouldn't form a question admitting an answer at present, as $(\infty,3)$-categories, symmetric monoidal $(\infty,3)$-categories, and also Gray tensor products of $(\infty,2)$-categories are all currently developing subjects.)
