  Let F be a right Hilbert C*-module over a C*-algebra B, and suppose that F is
equipped with a left action, by compact operators, of a second C*-algebra A.
Tensor product with F gives a functor from Hilbert C*-modules over A to Hilbert
C*-modules over B. We prove that under certain conditions (which are always
satisfied if, for instance, A is nuclear), the image of this functor can be
described in terms of coactions of a certain coalgebra canonically associated
to F. We then discuss several examples that fit into this framework: parabolic
induction of tempered group representations; Hermitian connections on Hilbert
C*-modules; Fourier (co)algebras of compact groups; and the maximal C*-dilation
of operator modules over non-self-adjoint operator algebras.
