Lemma 97.12.3. In the situation of Lemma 97.12.1 assume
$\mathcal{Y}' = \mathcal{Y}$ and $H = \text{id}_\mathcal {Y}$,
$G$ is representable by algebraic spaces and étale.
Then $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces and étale. If $G$ is also surjective, then $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is surjective.
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
is representable by an algebraic space étale over $U$. An object of this over $U'$ corresponds to an object $x'$ in the fibre category of $\mathcal{X}'$ over $Z_{U'}$ such that $G(x') \cong x|_{Z_{U'}}$. By assumption the $2$-fibre product
is representable by an algebraic space $W$ such that the projection $W \to Z$ is étale. Then (97.12.3.1) is representable by the algebraic space $F$ parametrizing sections of $W \to Z$ over $U$ introduced in Lemma 97.9.2. Since $F \to U$ is étale we conclude that $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces and étale. Finally, if $\mathcal{X}' \to \mathcal{X}$ is surjective also, then $W \to Z$ is surjective, and hence $F \to U$ is surjective by Lemma 97.9.1. Thus in this case $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}) \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is also surjective. $\square$
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