Lemma 81.4.7. Let $S$ be a scheme. Let $f : Y' \to Y$ be a proper morphism of algebraic spaces over $S$. Let $i : Z \to Y$ be a closed immersion. Set $E = Z \times _ Y Y'$. Picture
\[ \xymatrix{ E \ar[d]_ g \ar[r]_ j & Y' \ar[d]^ f \\ Z \ar[r]^ i & Y } \]
If $f$ is an isomorphism over $Y \setminus Z$, then the functor
\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale}) \times _{\mathop{\mathit{Sh}}\nolimits (E_{\acute{e}tale})} \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \]
is an equivalence of categories.
Proof.
Observe that $X = Y' \coprod Z \to Y$ is a proper surjective morphism. Thus it suffice to construct an equivalence of categories
\[ \mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale}) \times _{\mathop{\mathit{Sh}}\nolimits (E_{\acute{e}tale})} \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]
compatible with pullback functors from $Y$ because then we can use Lemma 81.4.5 to conclude. Thus let $(\mathcal{G}', \mathcal{G}, \alpha )$ be an object of $\mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale}) \times _{\mathop{\mathit{Sh}}\nolimits (E_{\acute{e}tale})} \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale})$ with notation as in Categories, Example 4.31.3. Then we can consider the sheaf $\mathcal{F}$ on $X$ defined by taking $\mathcal{G}'$ on the summand $Y'$ and $\mathcal{G}$ on the summand $Z$. We have
\[ X \times _ Y X = Y' \times _ Y Y' \amalg Y' \times _ Y Z \amalg Z \times _ Y Y' \amalg Z \times _ Y Z = Y' \times _ Y Y' \amalg E \amalg E \amalg Z \]
The isomorphisms of the two pullbacks of $\mathcal{F}$ to this algebraic space are obvious over the summands $E$, $E$, $Z$. The interesting part of the proof is to find an isomorphism $\text{pr}_{0, small}^{-1}\mathcal{G}' \to \text{pr}_{1, small}^{-1}\mathcal{G}'$ over $Y' \times _ Y Y'$ satisfying the cocycle condition. However, our assumption that $Y' \to Y$ is an isomorphism over $Y \setminus Z$ implies that
\[ h : Y \coprod E \times _ Z E \longrightarrow Y' \times _ Y Y' \]
is a surjective proper morphism. (It is in fact a finite morphism as it is the disjoint union of two closed immersions.) Hence it suffices to construct an isomorphism of the pullbacks of $\text{pr}_{0, small}^{-1}\mathcal{G}'$and $\text{pr}_{1, small}^{-1}\mathcal{G}'$ by $h_{small}$ satisfying a certain cocycle condition. For the diagonal, it is clear how to do this. And for the pullback to $E \times _ Z E$ we use that both sheaves pull back to the pullback of $\mathcal{G}$ by the morphism $E \times _ Z E \to Z$. We omit the details.
$\square$
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