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