Lemma 87.21.7. Let S be a scheme. Let P be a local property of morphisms of \textit{WAdm}^{count} which is stable under base change. Let f : X \to Y and g : Z \to Y be morphisms of locally countably indexed formal algebraic spaces over S. If f satisfies the equivalent conditions of Lemma 87.21.3 then so does \text{pr}_2 : X \times _ Y Z \to Z.
Proof. Choose a covering \{ Y_ j \to Y\} as in Definition 87.11.1. For each j choose a covering \{ X_{ji} \to Y_ j \times _ Y X\} as in Definition 87.11.1. For each j choose a covering \{ Z_{jk} \to Y_ j \times _ Y Z\} as in Definition 87.11.1. Observe that X_{ji} \times _{Y_ j} Z_{jk} is an affine formal algebraic space which is countably indexed, see Lemma 87.20.10. Then we see that
\{ X_{ji} \times _{Y_ j} Z_{jk} \to X \times _ Y Z\}
is a covering as in Definition 87.11.1. Moreover, the morphisms X_{ji} \times _{Y_ j} Z_{jk} \to Z factor through Z_{jk}. By assumption we know that X_{ji} \to Y_ j corresponds to a morphism B_ j \to A_{ji} of \text{WAdm}^{count} having property P. The morphisms Z_{jk} \to Y_ j correspond to morphisms B_ j \to C_{jk} in \text{WAdm}^{count}. Since X_{ji} \times _{Y_ j} Z_{jk} = \text{Spf}(A_{ji} \widehat{\otimes }_{B_ j} C_{jk}) by Lemma 87.16.4 we see that it suffices to show that C_{jk} \to A_{ji} \widehat{\otimes }_{B_ j} C_{jk} has property P which is exactly what the condition that P is stable under base change guarantees. \square
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