Free Doubly-Infinitary Distributive Categories are Cartesian Closed

We investigate categories in which products distribute over coproducts, a structure we call doubly-infinitary distributive categories. Through a range of examples, we explore how this notion relates to established concepts such as extensivity, infinitary distributivity, and cartesian closedness. We show that doubly-infinitary distributivity strictly strengthens the classical notion of infinitary distributivity. Moreover, we prove that free doubly-infinitary distributive categories are cartesian closed, unlike free distributive categories. The paper concludes with observations on non-canonical isomorphisms, alongside open questions and directions for future research.