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We consider two calculi of pseudodifferential operators on manifolds withfibered boundary: Mazzeo's edge calculus, which has as local model theoperators associated to products of closed manifolds with asymptoticallyhyperbolic spaces, and the phi calculus of Mazzeo and the second author, whichis similarly modeled on products of closed manifolds with asymptoticallyEuclidean spaces. We construct an adiabatic calculus of operators interpolatingbetween them, and use this to compute the `smooth' K-theory groups of the edgecalculus, determine the existence of Fredholm quantizations of ellipticsymbols, and establish a families index theorem in K-theory.

Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary