Strictification tensor product of 2-categories

Abstract

Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic constructions of a 2-category $\mathcal{C}\boxtimes\mathcal{D}$ satisfying $\textrm{Lax}(\mathcal{C},\textrm{Lax}(\mathcal{D},\mathcal{E})) \cong [\mathcal{C}\boxtimes \mathcal{D},\mathcal{E}]_\mathrm{lnt}$, hence generalising the case of the free distributive law $1\boxtimes 1$. We also discuss dual constructions.