## 78.3 Useful diagrams

We briefly restate the results of Groupoids in Spaces, Lemmas 77.11.4 and 77.11.5 for easy reference in this chapter. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. In the commutative diagram

the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.

The diagram

is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.

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