Starting from an exact correspondence between linear approximations and non-idempotent intersection types, we develop a general framework for building systems of intersection types characterizing normalization properties. We show how this construction, which uses in a fundamental way Melliès and Zeilberger's ``type systems as functors'' viewpoint, allows us to recover equivalent versions of every well known intersection type system (including Coppo and Dezani's original system, as well as its non-idempotent variants independently introduced by Gardner and de Carvalho). We also show how new systems of intersection types may be built almost automatically in this way.