Lemma 115.20.1. Notation and assumptions as above. Let $p : X_ A \to X$ denote the projection. Given $A'$ denote $p' : X_{A'} \to X$ the projection. The functor $p'_*$ induces an equivalence of categories between

the category $\textit{Lift}(\mathcal{F}, A')$, and

the category $\textit{Lift}(p_*\mathcal{F}, A')$.

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