  For Cuntz-Pimsner algebras of bi-Hilbertian bimodules of finite
Jones-Watatani index satisfying some side conditions, we give an explicit
isomorphism between the $K$-theory exact sequences of the mapping cone of the
inclusion of the coefficient algebra into a Cuntz-Pimsner algebra, and the
Cuntz-Pimsner exact sequence. In the process we extend some results by the
second author and collaborators from finite projective bimodules to finite
index bimodules, and also clarify some aspects of Pimsner's `extension of
scalars' construction.
