Lemma 10.154.12. Let $R \to S$ be a local map of local rings. Choose separable algebraic closures $R/\mathfrak m_ R \subset \kappa _1^{sep}$ and $S/\mathfrak m_ S \subset \kappa _2^{sep}$. Let $R \to R^{sh}$ and $S \to S^{sh}$ be the corresponding strict henselizations. Given any commutative diagram

There exists a unique local ring map $R^{sh} \to S^{sh}$ fitting into the commutative diagram

and inducing $\phi $ on the residue fields of $R^{sh}$ and $S^{sh}$.

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