Lemma 63.6.5. Consider a cartesian square

of schemes with $f$ locally quasi-finite. For any abelian sheaf $\mathcal{F}$ on $Y'_{\acute{e}tale}$ we have $(g')_*(f')^!\mathcal{F} = f^!g_*\mathcal{F}$.

Lemma 63.6.5. Consider a cartesian square

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

of schemes with $f$ locally quasi-finite. For any abelian sheaf $\mathcal{F}$ on $Y'_{\acute{e}tale}$ we have $(g')_*(f')^!\mathcal{F} = f^!g_*\mathcal{F}$.

**Proof.**
By uniqueness of adjoint functors, this follows from the corresponding (dual) statement for the functors $f_!$. See Lemma 63.4.10.
$\square$

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