Tensor products, internal homs, and model structures in two dimensional category theory

In this paper, we introduce a new symmetric monoidal structure on $\mathbf{Cat}$, called the \emph{graph tensor product}, with unit given by the terminal category. This tensor product falls in the middle of a factorization between the funny tensor product and the Cartesian product, giving a factorization connecting these two classical monoidal structures. We extend this construction to a symmetric monoidal structure on $2\mathbf{Cat}$, again with unit $D^0$, which provides an analogous factorization between the funny tensor product and the Cartesian product of $2$-categories. Using the $(\mathrm{bo},\mathrm{lff})$ factorization system on $2\mathbf{Cat}$, we construct a new symmetric monoidal closed model structure on $2\mathbf{Cat}$ whose tensor product restricts to the Cartesian product on the subcategory of flexible $2$-categories. Finally, we prove that this symmetric monoidal model structure fits into a square of weak symmetric monoidal Quillen equivalences relating the Gray tensor product and the flexible tensor product.