Lemma 37.76.2. The composition of two completely decomposed morphisms of schemes is completely decomposed. If $\{ f_ i : X_ i \to Y\} _{i \in I}$ is completely decomposed and for each $i$ we have a family $\{ X_{ij} \to X_ i\} _{j \in J_ i}$ which is completely decomposed, then the family $\{ X_{ij} \to Y\} _{i \in I, j \in J_ i}$ is completely decomposed.

**Proof.**
Omitted.
$\square$

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