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