Functoriality of Cuntz–Pimsner correspondence maps

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