Lemma 73.14.2. Let $S$ be a scheme. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}\} $. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\tau $ local on the source. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $a : X' \to X$ which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. smooth, resp. étale, the composition $f \circ a : X' \to Y$ has property $\mathcal{P}$.

**Proof.**
This is true because we can fit $X' \to X$ into a family of morphisms which forms a $\tau $-covering.
$\square$

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