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_op(B,Lax_op(C,D)). The collation of a distributive law into its associated bifunctor extends to a 2-functor into Lax_op(B x 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.
Keywords: morphism of bicategories, triple, braiding, curry, exponential
2020 MSC: 18D05, 18C15
Theory and Applications of Categories, Vol. 37, 2021, No. 34, pp 1149-1175.