Lemma 59.100.3. In Lemma 59.100.2 if f is proper, then we have a_ Y^{-1} \circ f_{small, *} = f_{big, fppf, *} \circ a_ X^{-1}.
Proof. You can prove this by repeating the proof of Lemma 59.99.5 part (1); we will instead deduce the result from this. As \epsilon _{Y, *} is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma 59.100.1 shows that \epsilon _{Y, *} \circ a_ Y^{-1} = \pi _ Y^{-1} and similarly for X. To show that the canonical map a_ Y^{-1}f_{small, *}\mathcal{F} \to f_{big, fppf, *}a_ X^{-1}\mathcal{F} is an isomorphism, it suffices to show that
\begin{align*} \pi _ Y^{-1}f_{small, *}\mathcal{F} & = \epsilon _{Y, *}a_ Y^{-1}f_{small, *}\mathcal{F} \\ & \to \epsilon _{Y, *}f_{big, fppf, *}a_ X^{-1}\mathcal{F} \\ & = f_{big, {\acute{e}tale}, *} \epsilon _{X, *}a_ X^{-1}\mathcal{F} \\ & = f_{big, {\acute{e}tale}, *}\pi _ X^{-1}\mathcal{F} \end{align*}
is an isomorphism. This is part (1) of Lemma 59.99.5. \square
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