  We consider two calculi of pseudodifferential operators on manifolds with
fibered boundary: Mazzeo's edge calculus, which has as local model the
operators associated to products of closed manifolds with asymptotically
hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which
is similarly modeled on products of closed manifolds with asymptotically
Euclidean spaces. We construct an adiabatic calculus of operators interpolating
between them, and use this to compute the `smooth' K-theory groups of the edge
calculus, determine the existence of Fredholm quantizations of elliptic
symbols, and establish a families index theorem in K-theory.
