Bifunctor Theorem and strictification tensor product for double categories with lax double functors

Abstract: We introduce a candidate for the inner hom for $Dbl^{st}_{lx}$, the category of strict double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a pair of lax double functors with four 2-cells resembling distributive laws. We extend this characterization to a 2-category isomorphism $q\x\Lax_{hop}(\Aa\times\Bb,\Cc) \iso \Lax_{hop}(\Aa, \llbracket\Bb,\Cc\rrbracket)$. We show that instead of a Gray monoidal product in $Dbl^{st}_{lx}$ we obtain a product that in a sense strictifies lax double quasi-functors. We prove a bifunctor theorem by which certain type of lax double quasi-functors give rise to lax double functors on the Cartesian product, extend it to a 2-functor $q\x\Lax_{hop}^{ns}(\Aa\times\Bb,\Cc)\to\Lax_{hop}(\Aa\times\Bb,\Cc)$ and show how it restricts to a biequivalence. The (un)currying 2-functors are studied. We prove that a lax double functor from the trivial double category is a monad in the codomain double category, and show that our above 2-functor in the form $q\x\Lax_{hop}(\times *,\Dd)\to\Lax_{hop}(,\Dd)$ recovers the specification $\Comp(\HH(\Dd)):\Mnd\Mnd(\HH(\Dd))\to\Mnd(\HH(\Dd))$ of the natural transformation $\Comp$ introduced by Street, where $\HH(\Dd)$ is the horizontal 2-category of $\Dd$.