The Stacks project

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 étale 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

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 étale $R$-algebras. Hence the uniqueness of Lemma 10.154.7 shows that $R^{sh}$ is the strict henselization.