Passing $C^*$-correspondence Relations to the Cuntz-Pimsner algebras

We construct a functor that maps $C^*$-correspondences to their Cuntz-Pimsner algebras. The objects in our domain category are $C^*$-correspondences, and the morphisms are the isomorphism classes of $C^*$-correspondences satisfying certain conditions. Applications include a generalization of the well-known result of Muhly and Solel: Morita equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras; as well as a generalization of the result of Kakariadis and Katsoulis: Regular shift equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras.