Lemma 15.12.3. The functor of Lemma 15.12.1 associates to a local ring $(A, \mathfrak m)$ its henselization.

** Compatibility henselization of pairs and of local rings. **

**Proof.**
Let $(A^ h, \mathfrak m^ h)$ be the henselization of the pair $(A, \mathfrak m)$ constructed in Lemma 15.12.1. Then $\mathfrak m^ h = \mathfrak m A^ h$ is a maximal ideal by Lemma 15.12.2 and since it is contained in the Jacobson radical, we conclude $A^ h$ is local with maximal ideal $\mathfrak m^ h$. Having said this there are two ways to finish the proof.

First proof: observe that the construction in the proof of Algebra, Lemma 10.154.1 as a colimit is the same as the colimit used to construct $A^ h$ in Lemma 15.12.1. Second proof: Both the henselization $A \to S$ and $A \to A^ h$ of Lemma 15.12.1 are local ring homomorphisms, both $S$ and $A^ h$ are filtered colimits of étale $A$-algebras, both $S$ and $A^ h$ are henselian local rings, and both $S$ and $A^ h$ have residue fields equal to $\kappa (\mathfrak m)$ (by Lemma 15.12.2 for the second case). Hence they are canonically isomorphic by Algebra, Lemma 10.153.6. $\square$

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