Lemma 35.26.2. Let \tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} . Let \mathcal{P} be a property of morphisms 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. étale, resp. an open immersion, 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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