Strictification tensor product of 2-categories

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