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Remark 10.155.4. We can also construct $R^{sh}$ from $R^ h$. Namely, for any finite separable subextension $\kappa ^{sep}/\kappa '/\kappa $ there exists a unique (up to unique isomorphism) finite ├ętale local ring extension $R^ h \subset R^ h(\kappa ')$ whose residue field extension reproduces the given extension, see Lemma 10.153.7. Hence we can set

\[ R^{sh} = \bigcup \nolimits _{\kappa \subset \kappa ' \subset \kappa ^{sep}} R^ h(\kappa ') \]

The arrows in this system, compatible with the arrows on the level of residue fields, exist by Lemma 10.153.7. This will produce a henselian local ring by Lemma 10.154.8 since each of the rings $R^ h(\kappa ')$ is henselian by Lemma 10.153.4. By construction the residue field extension induced by $R^ h \to R^{sh}$ is the field extension $\kappa ^{sep}/\kappa $. Hence $R^{sh}$ so constructed is strictly henselian. By Lemma 10.154.2 the $R$-algebra $R^{sh}$ is a colimit of ├ętale $R$-algebras. Hence the uniqueness of Lemma 10.154.7 shows that $R^{sh}$ is the strict henselization.

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