  We introduce positive correspondences as right C*-modules with left actions
given by completely positive maps. Positive correspondences form a
semi-category that contains the C*-correspondence (Enchilada) category as a
"retract". Kasparov's KSGNS construction provides a semi-functor from this
semi-category onto the C*-correspondence category. The need for left actions by
completely positive maps appears naturally when we consider morphisms between
Cuntz-Pimsner algebras, and we describe classes of examples arising from
projections on C*-correspondences and Fock spaces, as well as examples from
conjugation by bi-Hilbertian bimodules of finite index.
