Functoriality of Cuntz–Pimsner correspondence maps

We show that the passage from a C∗C∗-correspondence to its Cuntz–Pimsner C∗C∗-algebra gives a functor on a category of C∗C∗-correspondences with appropriately defined morphisms. Applications involving topological graph C∗C∗-algebras are discussed, and an application to crossed-product correspondences is presented in detail.