Lemma 87.21.16. Let S be a scheme. Let P be a local property of morphisms of \textit{WAdm}^{count} which has the cancellation property. Let f : X \to Y and g : Y \to Z be morphisms of locally countably indexed formal algebraic spaces over S. If g \circ f and g satisfies the equivalent conditions of Lemma 87.21.3 then so does f : X \to Y.
Proof. Choose a covering \{ Z_ k \to Z\} as in Definition 87.11.1. For each k choose a covering \{ Y_{kj} \to Z_ k \times _ Z Y\} as in Definition 87.11.1. For each k and j choose a covering \{ X_{kji} \to Y_{kj} \times _ Y X\} as in Definition 87.11.1. Let X_{kji} \to Y_{jk} and Y_{jk} \to Z_ k correspond to arrows B_{kj} \to A_{kji} and C_ k \to B_{kj} of \text{WAdm}^{count}. If g \circ f and g satisfies the equivalent conditions of Lemma 87.21.3 then C_ k \to B_{kj} and C_ k \to A_{kji} satisfy P. Hence B_{kj} \to A_{kji} does too and we conclude. \square
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