Change of weights operations for triangulated $$(\varphi ,\Gamma )$$-modules

Abstract Wu has shown the existence of “change of weights” operation on $$(\varphi ,\Gamma )$$ ( 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.