Lemma 87.37.4. Let S be a scheme. Let f : X' \to X be a morphism of formal algebraic spaces over S. Let T \subset |X_{red}| be a closed subset and let T' = |f_{red}|^{-1}(T) \subset |X'_{red}|. Then
is a cartesian diagram of formal algebraic spaces over S.
Proof. Namely, observe that the horizontal arrows are monomorphisms by construction. Thus it suffices to show that a morphism g : U \to X' from a scheme U defines a point of X'_{/T} if and only if f \circ g defines a point of X_{/T}. In other words, we have to show that g(U) is contained in T' \subset |X'_{red}| if and only if (f \circ g)(U) is contained in T \subset |X_{red}|. This follows immediately from our choice of T' as the inverse image of T. \square
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