Lemma 84.6.7. In Lemma 84.6.4 assume $f$ is flat, locally of finite presentation, and surjective. Then the functor
\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \left\{ (\mathcal{G}, \mathcal{H}, \alpha ) \middle | \begin{matrix} \mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}),\ \mathcal{H} \in \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_{fppf}),
\\ \alpha : a_ X^{-1}\mathcal{G} \to f_{big, fppf}^{-1}\mathcal{H} \text{ an isomorphism}
\end{matrix} \right\} \]
sending $\mathcal{F}$ to $(f_{small}^{-1}\mathcal{F}, a_ Y^{-1}\mathcal{F}, can)$ is an equivalence.
Proof.
The functor $a_ X^{-1}$ is fully faithful (as $a_{X, *}a_ X^{-1} = \text{id}$ by Lemma 84.6.1). Hence the forgetful functor $(\mathcal{G}, \mathcal{H}, \alpha ) \mapsto \mathcal{H}$ identifies the category of triples with a full subcategory of $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_{fppf})$. Moreover, the functor $a_ Y^{-1}$ is fully faithful, hence the functor in the lemma is fully faithful as well.
Suppose that we have an étale covering $\{ Y_ i \to Y\} $. Let $f_ i : X_ i \to Y_ i$ be the base change of $f$. Denote $f_{ij} = f_ i \times f_ j : X_ i \times _ X X_ j \to Y_ i \times _ Y Y_ j$. Claim: if the lemma is true for $f_ i$ and $f_{ij}$ for all $i, j$, then the lemma is true for $f$. To see this, note that the given étale covering determines an étale covering of the final object in each of the four sites $Y_{\acute{e}tale}, X_{\acute{e}tale}, (\mathit{Sch}/Y)_{fppf}, (\mathit{Sch}/X)_{fppf}$. Thus the category of sheaves is equivalent to the category of glueing data for this covering (Sites, Lemma 7.26.5) in each of the four cases. A huge commutative diagram of categories then finishes the proof of the claim. We omit the details. The claim shows that we may work étale locally on $Y$. In particular, we may assume $Y$ is a scheme.
Assume $Y$ is a scheme. Choose a scheme $X'$ and a surjective étale morphism $s : X' \to X$. Set $f' = f \circ s : X' \to Y$ and observe that $f'$ is surjective, locally of finite presentation, and flat. Claim: if the lemma is true for $f'$, then it is true for $f$. Namely, given a triple $(\mathcal{G}, \mathcal{H}, \alpha )$ for $f$, we can pullback by $s$ to get a triple $(s_{small}^{-1}\mathcal{G}, \mathcal{H}, s_{big, fppf}^{-1}\alpha )$ for $f'$. A solution for this triple gives a sheaf $\mathcal{F}$ on $Y_{\acute{e}tale}$ with $a_ Y^{-1}\mathcal{F} = \mathcal{H}$. By the first paragraph of the proof this means the triple is in the essential image. This reduces us to the case where both $X$ and $Y$ are schemes. This case follows from Étale Cohomology, Lemma 59.100.4 via the discussion in Section 84.3 and in particular Lemma 84.3.1.
$\square$
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