The Stacks project

Proof. Observe that $h$ is representable by algebraic spaces by Morphisms of Stacks, Lemmas 101.45.3 and 101.45.1. Choose an affine scheme $U$ and a surjective, flat, finite, and finitely presented morphism $U \to \mathcal{X}$. Then $U' = \mathcal{X}' \times _\mathcal {X} U$ is an algebraic space with a finite (in particular affine) morphism $U' \to \mathcal{X}'$. By Lemma 106.13.3 we conclude that $U'$ is affine. Setting $R = U \times _\mathcal {X} U$ and $R' = U' \times _{\mathcal{X}'} U'$ we obtain groupoids $(U, R, s, t, c)$ and $(U', R', s', t', c')$ such that $\mathcal{X} = [U/R]$ and $\mathcal{X}' = [U'/R']$, see proof of Lemma 106.13.2. we see that the diagrams

are cartesian where $G$ and $G'$ are the stabilizer group schemes. This follows for the first two by transitivity of fibre products and for the last one this follows because it is the pullback of the isomorphism $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' \times _\mathcal {X} \mathcal{I}_\mathcal {X}$ (by the already used Morphisms of Stacks, Lemma 101.45.3). Recall that $M$, resp. $M'$ was constructed in Lemma 106.13.4 as the spectrum of the ring of $R$-invariant functions on $U$, resp. the ring of $R'$-invariant functions on $U'$. Thus $M' \to M$ is étale and $U' = M' \times _ M U$ by Groupoids, Lemma 39.23.7. It follows that $R' = M' \times _ M U$, in other words the groupoid $(U', R', s', t', c')$ is the base change of $(U, R, s, t, c)$ by $M' \to M$. This implies that the diagram in the lemma is cartesian by Quotients of Groupoids, Lemma 83.3.6. $\square$