Lemma 74.22.5. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Set $Y = \coprod _{i \in I} X_ i$. There is a canonical equivalence of categories
\[ \begin{matrix} \text{category of descent data }
\\ \text{relative to the family } \{ X_ i \to X\} _{i \in I}
\end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data}
\\ \text{ relative to } Y/X
\end{matrix} \]
which maps $(V_ i, \varphi _{ij})$ to $(V, \varphi )$ with $V = \coprod _{i\in I} V_ i$ and $\varphi = \coprod \varphi _{ij}$.
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