The Stacks project

Lemma 40.9.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$, and let $h = s \circ \text{pr}_1 : U' \times _{g, U, t} R \to U$. If $\mathcal{P}$ is a property of morphisms of schemes such that

1. $h$ has property $\mathcal{P}$, and

2. $\mathcal{P}$ is preserved under base change,

then $s', t'$ have property $\mathcal{P}$.