Change of weights operations for triangulated $(\phi ,\mathrm{\Gamma })$-modules

Zhixiang Wu has shown the existence of “change of weights” operation on $(\phi ,\mathrm{\Gamma })$-modules in families (arxiv:2405.16637). 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 Yiwen Ding (arXiv:2407.21237) in the non-critical crystabelline case.