The Stacks project

Lemma 35.32.6. Let $f : X' \to X$ be a morphism of schemes over a base scheme $S$. Assume there exists a morphism $g : X \to X'$ over $S$, for example if $f$ has a section. Then the pullback functor of Lemma 35.31.6 defines an equivalence of categories between the category of descent data relative to $X/S$ and $X'/S$.

Proof. Let $g : X \to X'$ be a morphism over $S$. Lemma 35.32.5 above shows that the functors $f^* \circ g^* = (g \circ f)^*$ and $g^* \circ f^* = (f \circ g)^*$ are isomorphic to the respective identity functors as desired. $\square$


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