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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