Lemma 87.21.12. Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which is stable under composition. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ and $g$ satisfies the equivalent conditions of Lemma 87.21.3 then so does $g \circ f : X \to Z$.
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. If $f$ and $g$ satisfies the equivalent conditions of Lemma 87.21.3 then $X_{kji} \to Y_{jk}$ and $Y_{jk} \to Z_ k$ correspond to arrows $B_{kj} \to A_{kji}$ and $C_ k \to B_{kj}$ of $\text{WAdm}^{count}$ having property $P$. Hence the compositions do too and we conclude. $\square$
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