Lemma 96.12.6. Let $F : \mathcal{X} \to \mathcal{Y}$ and $G : \mathcal{X}' \to \mathcal{X}$ be $1$-morphisms of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $G$ is representable by algebraic spaces, then the $1$-morphism

$\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}) \longrightarrow \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$

is representable by algebraic spaces.

Proof. Let $U$ be a scheme and let $\xi = (U, Z, y, x, \alpha )$ be an object of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ over $U$. We have to prove that the $2$-fibre product

96.12.6.1
$$\label{criteria-equation-to-show-again} (\mathit{Sch}/U)_{fppf} \times _{\xi , \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})} \mathcal{H}_ d(\mathcal{X}'/\mathcal{Y})$$

is representable by an algebraic space étale over $U$. An object of this over $a : U' \to U$ corresponds to an object $x'$ of $\mathcal{X}'$ over $U' \times _{a, U} Z$ such that $G(x') \cong x|_{U' \times _{a, U} Z}$. By assumption the $2$-fibre product

$(\mathit{Sch}/Z)_{fppf} \times _{x, \mathcal{X}} \mathcal{X}'$

is representable by an algebraic space $X$ over $Z$. It follows that (96.12.6.1) is representable by $\text{Res}_{Z/U}(X)$, which is an algebraic space by Proposition 96.11.5. $\square$

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