Symmetric Strong Monoidality With Respect to Products. The connected components functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\pi _{0},\pi ^{\times }_{0},\pi ^{\times }_{0|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Cats},\times ,\mathsf{pt}\webright ) \to \webleft (\mathsf{Sets},\times ,\text{pt}\webright ), \]
being equipped with isomorphisms
\[ \begin{gathered} \pi ^{\times }_{0|\mathcal{C},\mathcal{D}} \colon \pi _{0}\webleft (\mathcal{C}\webright )\times \pi _{0}\webleft (\mathcal{D}\webright ) \xrightarrow {\cong }\pi _{0}\webleft (\mathcal{C}\times \mathcal{D}\webright ),\\ \pi ^{\times }_{0|\mathbb {1}} \colon \text{pt}\xrightarrow {\cong }\pi _{0}\webleft (\mathsf{pt}\webright ), \end{gathered} \]
natural in $\mathcal{C},\mathcal{D}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.