Lemma 35.35.8. Let f : X \to X' be a morphism of schemes over a base scheme S. Assume \{ X \to S\} is an fpqc covering (for example if f is surjective, flat and quasi-compact). Then the pullback functor of Lemma 35.34.6 is a fully faithful functor from the category of descent data relative to X'/S to the category of descent data relative to X/S.
Proof. We may factor X \to X' as X \to X \times _ S X' \to X'. The first morphism has a section, hence induces an equivalence of categories of descent data by Lemma 35.35.6. The second morphism is an fpqc covering hence induces a fully faithful functor by Lemma 35.35.4. \square
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