The Stacks project

Lemma 35.34.5. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i \in I}$ be a family of morphisms with target $S$. Set $X = \coprod _{i \in I} X_ i$, and consider it as an $S$-scheme. There is a canonical equivalence of categories

\[ \begin{matrix} \text{category of descent data } \\ \text{relative to the family } \{ X_ i \to S\} _{i \in I} \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/S \end{matrix} \]

which maps $(V_ i, \varphi _{ij})$ to $(V, \varphi )$ with $V = \coprod _{i\in I} V_ i$ and $\varphi = \coprod \varphi _{ij}$.

Proof. Observe that $X \times _ S X = \coprod _{ij} X_ i \times _ S X_ j$ and similarly for higher fibre products. Giving a morphism $V \to X$ is exactly the same as giving a family $V_ i \to X_ i$. And giving a descent datum $\varphi $ is exactly the same as giving a family $\varphi _{ij}$. $\square$

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