Lemma 65.16.5. Let \mathit{Sch}_{fppf} be a big fppf site. Let S \to S' be a morphism of this site. Let F be an algebraic space over S. Let T be a scheme over S and let f : T \to F be a morphism over S. Let f' : T' \to F' be the morphism over S' we get from f by applying the equivalence of categories described in Lemma 65.16.3. For any property \mathcal{P} as in Definition 65.5.1 we have \mathcal{P}(f') \Leftrightarrow \mathcal{P}(f).
Proof. Suppose that U is a scheme over S, and U \to F is a surjective étale morphism. Denote U' the scheme U viewed as a scheme over S'. In Lemma 65.16.1 we have seen that U' \to F' is surjective étale. Since
j(T \times _{f, F} U) = T' \times _{f', F'} U'
the morphism of schemes T \times _{f, F} U \to U is identified with the morphism of schemes T' \times _{f', F'} U' \to U'. It is the same morphism, just viewed over different base schemes. Hence the lemma follows from Lemma 65.11.4. \square
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