Lemma 87.38.2. Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace and let $Z' = f^{-1}(Z) = X' \times _ X Z$. Then
is a cartesian diagram of sheaves. In particular, the morphism $(X')^\wedge _{Z'} \to X^\wedge _ Z$ is representable by algebraic spaces.
Proof. Namely, suppose that $Y \to X$ is a morphism from a scheme into $X$ such that $Y \to X$ factors through $Z$. Then $Y \times _ X X' \to X$ is a morphism of algebraic spaces such that $Y \times _ X X' \to X'$ factors through $Z'$. Since $Z'_ n = X' \times _ X Z_ n$ for all $n \geq 1$ the same is true for the infinitesimal neighbourhoods. Hence the cartesian square of functors follows from the formulas $X^\wedge _ Z = \mathop{\mathrm{colim}}\nolimits Z_ n$ and $(X')^\wedge _{Z'} = \mathop{\mathrm{colim}}\nolimits Z'_ n$. $\square$
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