L-Fuzzifying topologies as L-topologies

It is proved in this paper that for every completely distributive lattice L with a locally multiplicative way below relation the category of L-fuzzifying topological spaces can be embedded in the category of L-topological spaces (stratified Chang-Goguen topological spaces) as a simultaneously bireflective and bicoreflective full subcategory. Several equivalent descriptions of this subcategory are given.