Lemma 35.32.3. In the situation of Lemma 35.31.6 assume that $f : X' \to X$ is surjective and flat. Then the pullback functor is faithful.

Proof. Let $(V_ i, \varphi _ i)$, $i = 1, 2$ be descent data for $X \to S$. Let $\alpha , \beta : V_1 \to V_2$ be morphisms of descent data. Suppose that $f^*\alpha = f^*\beta$. Our task is to show that $\alpha = \beta$. Note that $\alpha$, $\beta$ are morphisms of schemes over $X$, and that $f^*\alpha$, $f^*\beta$ are simply the base changes of $\alpha$, $\beta$ to morphisms over $X'$. Hence the lemma follows from Lemma 35.32.2. $\square$

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