On families of filtered (φ,N)-modules

In this paper, as a generalization of Berger's construction, we give a functor from the category of families of filtered (φ,N)-modules (with certain condition) to the category of families of (φ, Γ)-modules. Combining this with Kedlaya and Liu's theorem we show that, when the base is a reduced affinoid space, every family of weakly admissible filtered (φ,N)-modules can locally be converted into a family of semistable Galois representations.