Change of weights operations for triangulated (φ ,Γ )-modules

Wu has shown the existence of “change of weights” operation on (φ ,Γ ) ( φ , Γ ) -modules in families, [36, Prop. 3.16]. We interpret it in the trianguline case as pullbacks with a discussion on related stacks. Finally, we prove that it intertwines well with translation functors via a 1-1 correspondence defined by Ding [20] in the non-critical crystabelline case.