Lemma 59.87.2. Consider a cartesian diagram of schemes

where $f$ is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ on $T_{\acute{e}tale}$.

Lemma 59.87.2. Consider a cartesian diagram of schemes

\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]

where $f$ is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ on $T_{\acute{e}tale}$.

**Proof.**
Combine Lemma 59.87.1 with More on Morphisms, Lemma 37.46.3.
$\square$

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