Lemma 35.35.7. Let f : X \to X' be a morphism of schemes over a base scheme S. Assume X \to S is surjective and flat. Then the pullback functor of Lemma 35.34.6 is a 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 surjective and flat, hence induces a faithful functor by Lemma 35.35.3. \square
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