  Given a local Haag-Kastler net of von Neumann algebras and one of its scaling
limit states, we introduce a variant of the notion of asymptotic morphism by
Connes and Higson, and we show that the unitary equivalence classes of
(localized) morphisms of the scaling limit theory of the original net are in
bijection with classes of suitable pairs of such asymptotic morphisms. In the
process, we also show that the quasi-local C*-algebras of two nets are
isomorphic under very general hypotheses, and we construct an extension of the
scaling algebra whose representation on the scaling limit Hilbert space
contains the local von Neumann algebras. We also study the relation between our
asymptotic morphisms and superselection sectors preserved in the scaling limit.
