Lemma 10.155.5. Let R \to S be a local map of local rings. Let S \to S^ h be the henselization. Let R \to A be an étale ring map and let \mathfrak q be a prime of A lying over \mathfrak m_ R such that R/\mathfrak m_ R \cong \kappa (\mathfrak q). Then there exists a unique morphism of rings f : A \to S^ h fitting into the commutative diagram
\xymatrix{ A \ar[r]_ f & S^ h \\ R \ar[u] \ar[r] & S \ar[u] }
such that f^{-1}(\mathfrak m_{S^ h}) = \mathfrak q.
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