Remark 59.75.2 (Yet another description of sp). Let $S$, $\overline{s}$, and $\overline{t}$ be as above. Another alternative is to use the unique morphism

$c : \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}) \longrightarrow \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$

over $S$ which is compatible with the given morphism $\overline{t} \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ and the morphism $\overline{t} \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{t, \overline{t}})$. The uniqueness and existence of the displayed arrow follows from Algebra, Lemma 10.154.6 applied to $\mathcal{O}_{S, s}$, $\mathcal{O}^{sh}_{S, \overline{t}}$, and $\mathcal{O}^{sh}_{S, \overline{s}} \to \kappa (\overline{t})$. We obtain

$sp : \mathcal{F}_{\overline{s}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), \mathcal{F}) \xrightarrow {\text{pullback by }c} \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}), \mathcal{F}) = \mathcal{F}_{\overline{t}}$

(with obvious notational conventions). In fact this procedure also works for objects $K$ in $D(S_{\acute{e}tale})$: the specialization map for $K$ is equal to the map

$sp : K_{\overline{s}} = R\Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), K) \xrightarrow {\text{pullback by }c} R\Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}), K) = K_{\overline{t}}$

The equality signs are valid as taking global sections over the striclty henselian schemes $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ and $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}})$ is exact (and the same as taking stalks at $\overline{s}$ and $\overline{t}$) and hence no subtleties related to the fact that $K$ may be unbounded arise.

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