Lemma 10.155.10. 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
\[ \xymatrix{ \kappa _1^{sep} \ar[r]_{\phi } & \kappa _2^{sep} \\ R/\mathfrak m_ R \ar[r]^{\varphi } \ar[u] & S/\mathfrak m_ S \ar[u] } \]
There exists a unique local ring map $R^{sh} \to S^{sh}$ fitting into the commutative diagram
\[ \xymatrix{ R^{sh} \ar[r]_ f & S^{sh} \\ R \ar[u] \ar[r] & S \ar[u] } \]
and inducing $\phi $ on the residue fields of $R^{sh}$ and $S^{sh}$.
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