The Stacks project

Lemma 75.9.2. Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary distinguished square of algebraic spaces over $S$.

1. If $V' \subset V$ and $U \subset U' \subset W$ are open subspaces and $W' = U' \cup f(V')$ then $(U' \subset W', f|_{V'} : V' \to W')$ is an elementary distinguished square.

2. If $p : W' \to W$ is a morphism of algebraic spaces, then $(p^{-1}(U) \subset W', V \times _ W W' \to W')$ is an elementary distinguished square.

3. If $S' \to S$ is a morphism of schemes, then $(S' \times _ S U \subset S' \times _ S W, S' \times _ S V \to S' \times _ S W)$ is an elementary distinguished square.