Lemma 58.16.3. Let $f : X \to S$ be a proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization of points of $S$ and let $sp : \pi _1(X_{\overline{s}'}) \to \pi _1(X_{\overline{s}})$ be a specialization map. Then there exists a strictly henselian valuation ring $R$ over $S$ with algebraically closed fraction field such that $sp$ is isomorphic to $sp_ R$ defined above.

**Proof.**
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization constructed using $\kappa (s) \to \kappa (\overline{s})$. Let $A \to \kappa (\overline{s}')$ be the map used to construct $sp$. Let $R \subset \kappa (\overline{s}')$ be a valuation ring with fraction field $\kappa (\overline{s}')$ dominating the image of $A$. See Algebra, Lemma 10.50.2. Observe that $R$ is strictly henselian for example by Lemma 58.12.2 and Algebra, Lemma 10.50.3. Then the lemma is clear.
$\square$

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