Lemma 87.15.2. Let S be a scheme. Let X, Y be formal algebraic spaces over S and let Z be a sheaf whose diagonal is representable by algebraic spaces. Let X \to Z and Y \to Z be maps of sheaves. Then X \times _ Z Y is a formal algebraic space.
Proof. Choose \{ X_ i \to X\} and \{ Y_ j \to Y\} as in Definition 87.11.1. Then \{ X_ i \times _ Z Y_ j \to X \times _ Z Y\} is a family of maps which are representable by algebraic spaces and étale. Thus Lemma 87.15.1 tells us it suffices to show that X \times _ Z Y is a formal algebraic space when X and Y are affine formal algebraic spaces.
Assume X and Y are affine formal algebraic spaces. Write X = \mathop{\mathrm{colim}}\nolimits X_\lambda and Y = \mathop{\mathrm{colim}}\nolimits Y_\mu as in Definition 87.9.1. Then X \times _ Z Y = \mathop{\mathrm{colim}}\nolimits X_\lambda \times _ Z Y_\mu . Each X_\lambda \times _ Z Y_\mu is an algebraic space. For \lambda \leq \lambda ' and \mu \leq \mu ' the morphism
X_\lambda \times _ Z Y_\mu \to X_\lambda \times _ Z Y_{\mu '} \to X_{\lambda '} \times _ Z Y_{\mu '}
is a thickening as a composition of base changes of thickenings. Thus we conclude by applying Lemma 87.13.1. \square
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