Lemma 59.87.4. Consider a cartesian diagram of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
Assume that
$f$ is flat and open,
the residue fields of $S$ are separably algebraically closed,
given an étale morphism $U \to X$ with $U$ affine we can write $U$ as a finite disjoint union of open subschemes of $X$ (for example if $X$ is a normal integral scheme with separably closed function field),
any nonempty open of a fibre $X_ s$ of $f$ is connected (for example if $X_ s$ is irreducible or empty).
Then for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$ we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.
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