We introduce a candidate for the inner hom for the category of 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 double category isomorphism. We show that instead of a Gray monoidal product we obtain a product that in a sense strictifies lax double quasi-functors. We explain why laxity of double functors hinders our candidate for the inner hom from making the category of double categories and lax double functors a closed and enriched category over 2-categories (or double categories). We prove a bifunctor theorem by which certain type of lax double quasi-functors give rise to lax double functors on the Cartesian product. We extend this theorem to a double functor between double categories and show how it restricts to a double equivalence. The (un)currying double 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 double functor recovers the specification in that double category of the composition natural transformation on the monad functor.
Keywords: bicategories, double categories, Gray monoidal product
2020 MSC: 18N10
Theory and Applications of Categories, Vol. 39, 2023, No. 29, pp 824-873.