Lemma 35.32.5. Let $X \to S$ be a morphism of schemes. Let $f : X \to X$ be a selfmap of $X$ over $S$. In this case pullback by $f$ is isomorphic to the identity functor on the category of descent data relative to $X \to S$.

Proof. This is clear from Lemma 35.31.6 since it tells us that $f^* \cong \text{id}^*$. $\square$

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