Lemma 10.155.9. Let $\varphi : R \to S$ be a local map of local rings. Let $S/\mathfrak m_ S \subset \kappa ^{sep}$ be a separable algebraic closure. Let $S \to S^{sh}$ be the strict henselization of $S$ with respect to $S/\mathfrak m_ S \subset \kappa ^{sep}$. Let $R \to A$ be an étale ring map and let $\mathfrak q$ be a prime of $A$ lying over $\mathfrak m_ R$. Given any commutative diagram

there exists a unique morphism of rings $f : A \to S^{sh}$ fitting into the commutative diagram

such that $f^{-1}(\mathfrak m_{S^ h}) = \mathfrak q$ and the induced map $\kappa (\mathfrak q) \to \kappa ^{sep}$ is the given one.

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