2-dimensional bifunctor theorems and distributive laws

Abstract: In this paper we consider the conditions that need to be satisfied by two families of pseudofunctors with a common codomain for them to be collated into a bifunctor. We observe similarities between these conditions and distributive laws of monads before providing a unified framework from which both of these results may be inferred. We do this by proving a version of the bifunctor theorem for lax functors. We then show that these generalised distributive laws may be arranged into a 2-category Dist(B,C,D), which is equivalent to Lax(B,Lax(C,D)). The collation of a distributive law into its associated bifunctor extends to a 2-functor into Lax($B \times C$, D), which corresponds to uncurrying via the aforementioned equivalence. We also describe subcategories on which collation itself restricts to an equivalence. Finally, we exhibit a number of natural categorical constructions as special cases of our result.