Lemma 39.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. The construction of Lemma 39.21.2 determines an equivalence

$\begin{matrix} \text{category of groupoid schemes} \\ \text{cartesian over } (X, X \times _ Y X, \ldots ) \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/Y \end{matrix}$

Proof. This is clear from Lemma 39.21.2 and the definition of descent data on schemes in Descent, Definition 35.34.1. $\square$

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