Symmetric Strong Monoidality With Respect to Products. The core functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\mathsf{Core},\mathsf{Core}^{\times },\mathsf{Core}^{\times }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Cats},\times ,\mathsf{pt}\webright ) \to \webleft (\mathsf{Grpd},\times ,\mathsf{pt}\webright ) \]
being equipped with isomorphisms
\[ \begin{gathered} \mathsf{Core}^{\times }_{\mathcal{C},\mathcal{D}} \colon \mathsf{Core}\webleft (\mathcal{C}\webright )\times \mathsf{Core}\webleft (\mathcal{D}\webright ) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Core}\webleft (\mathcal{C}\times \mathcal{D}\webright ),\\ \mathsf{Core}^{\times }_{\mathbb {1}} \colon \mathsf{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Core}\webleft (\mathsf{pt}\webright ), \end{gathered} \]
natural in $\mathcal{C},\mathcal{D}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.