Lemma 88.27.1. In the situation above, let X_1 \to X be a morphism of algebraic spaces with X_1 locally Noetherian. Denote T_1 \subset |X_1| the inverse image of T and U_1 \subset X_1 the inverse image of U. We denote
\mathcal{C}_{X, T} the category whose objects are morphisms of algebraic spaces f : X' \to X which are locally of finite type and such that U' = f^{-1}U \to U is an isomorphism,
\mathcal{C}_{X_1, T_1} the category whose objects are morphisms of algebraic spaces f_1 : X_1' \to X_1 which are locally of finite type and such that f_1^{-1}U_1 \to U_1 is an isomorphism,
\mathcal{C}_{X_{/T}} the category whose objects are morphisms g : W \to X_{/T} of formal algebraic spaces with W locally Noetherian and g rig-étale,
\mathcal{C}_{X_{1, /T_1}} the category whose objects are morphisms g_1 : W_1 \to X_{1, /T_1} of formal algebraic spaces with W_1 locally Noetherian and g_1 rig-étale.
Then the diagram
is commutative where the horizontal arrows are given by (88.27.0.1) and the vertical arrows by base change along X_1 \to X and along X_{1, /T_1} \to X_{/T}.
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