Lemma 97.12.4. In the situation of Lemma 97.12.1. Assume that $G$, $H$ are 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 also $H$ is surjective and the induced functor $\mathcal{X}' \to \mathcal{Y}' \times _\mathcal {Y} \mathcal{X}$ is surjective, then $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is surjective.
Proof. Set $\mathcal{X}'' = \mathcal{Y}' \times _\mathcal {Y} \mathcal{X}$. By Lemma 97.4.1 the $1$-morphism $\mathcal{X}' \to \mathcal{X}''$ is representable by algebraic spaces and étale (in particular the condition in the second statement of the lemma that $\mathcal{X}' \to \mathcal{X}''$ be surjective makes sense). We obtain a $2$-commutative diagram
\[ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X}'' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} } \]
It follows from Lemma 97.12.2 that $\mathcal{H}_ d(\mathcal{X}''/\mathcal{Y}')$ is the base change of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ by $\mathcal{Y}' \to \mathcal{Y}$. In particular we see that $\mathcal{H}_ d(\mathcal{X}''/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces and étale, see Algebraic Stacks, Lemma 94.10.6. Moreover, it is also surjective if $H$ is. Hence if we can show that the result holds for the left square in the diagram, then we're done. In this way we reduce to the case where $\mathcal{Y}' = \mathcal{Y}$ which is the content of Lemma 97.12.3. $\square$
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