Remark 59.75.8. Let $f : X \to S$ be a morphism of schemes. Let $K \in D(X_{\acute{e}tale})$. Let $\overline{s}$ be a geometric point of $S$ and let $\overline{t}$ be a geometric point of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. Let $c$ be as in Remark 59.75.2. We can always make a commutative diagram

$\xymatrix{ (Rf_*K)_{\overline{s}} \ar[r] \ar[d]_{sp} & R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), K) \ar[r] \ar[d]_{(\text{id}_ X \times c)^{-1}} & R\Gamma (X_{\overline{s}}, K) \\ (Rf_*K)_{\overline{t}} \ar[r] & R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{t}}^{sh}), K) \ar[r] & R\Gamma (X_{\overline{t}}, K) }$

where the horizontal arrows are those of Remark 59.53.2. In general there won't be a vertical map on the right between the cohomologies of $K$ on the fibres fitting into this diagram, even in the case of Lemma 59.75.7.

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