Lemma 80.3.2. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{I} \to (\mathit{Sch}/S)_{fppf}$, $i \mapsto X_ i$ be a diagram of algebraic spaces over $B$. Let $(X, X_ i \to X)$ be a cocone for the diagram in the category of algebraic spaces over $B$ (Categories, Remark 4.14.5). If there exists a fpqc covering $\{ U_ a \to X\} _{a \in A}$ such that

for all $a \in A$ we have $U_ a = \mathop{\mathrm{colim}}\nolimits X_ i \times _ X U_ a$ in the category of algebraic spaces over $B$, and

for all $a, b \in A$ we have $U_ a \times _ X U_ b = \mathop{\mathrm{colim}}\nolimits X_ i \times _ X U_ a \times _ X U_ b$ in the category of algebraic spaces over $B$,

then $X = \mathop{\mathrm{colim}}\nolimits X_ i$ in the category of algebraic spaces over $B$.

## Comments (1)

Comment #7782 by Laurent Moret-Bailly on

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