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
\xymatrix{ (X')^\wedge _{Z'} \ar[r] \ar[d] & X' \ar[d]^ f \\ X^\wedge _ Z \ar[r] & X }
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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