The nerve $N(\mathsf{C})$ of a category $\mathsf{C}$ can be seen as a geometric realization of it (via n-simplices). This defines a functor $N: \mathsf{Cat} \rightarrow \mathsf{SSet}$ called nerve functor. The nerve functor $N: \mathsf{Cat} \rightarrow \mathsf{SSet}$ is fully and faithful. See porposition 3.11 in Ref 1.
The motivation is to generalize this construction to the case of symmetric monoidal categories. The attempt are in Ref 2.
The problem is how to show the functor from the category of small symmetric monoidal categories to the category of copresheaves on $\mathsf{Cau}$ defined in theorem 1.2.1 in Ref 2 is fully faithful. The key point is to show for any symmetric monoidal category $\mathsf{S}_1$ and $\mathsf{S}_2$, ${\rm Hom}(\mathsf{S}_1,\mathsf{S}_2)\cong{\rm Hom}(\mathcal{F}_{\mathsf{S}_1},\mathcal{F}_{\mathsf{S}_2})$.
I am currently unable to devise a proof or establish a concrete approach to this point. Hence, I am seeking assistance from anyone who might be able to shed light on this matter. Your guidance and insights would be greatly appreciated.
