The Stacks project

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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