Lemma 59.102.3. In Lemma 59.102.2 if $f$ is proper, then we have $a_ Y^{-1} \circ f_{small, *} = f_{big, ph, *} \circ a_ X^{-1}$.

**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 59.102.1 shows that $\epsilon _{Y, *} \circ a_ Y^{-1} = \pi _ Y^{-1}$ and similarly for $X$. To show that the canonical map $a_ Y^{-1}f_{small, *}\mathcal{F} \to f_{big, ph, *}a_ X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that

is an isomorphism. This is part (1) of Lemma 59.99.5. $\square$

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