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The extensive completion of a distributive category

A category with finite products and finite coproducts is said to be distributive if the canonical map A×B + A×C → A× (B + C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D → Dex preserving finite products and finite coproducts, for which Dex is extensive; that is, for all objects A and B the functor Dex/A × Dex/B → Dex/(A + B) is an equivalence of categories. As an application, we show that a distributive category D has a full distributive embedding into the product of an extensive category with products and a distributive preorder.