Lemma 10.154.6. Let $R$ be a ring. Given a commutative diagram of ring maps
where $S$, $S'$ are henselian local, $S$, $S'$ are filtered colimits of étale $R$-algebras, $K$ is a field and the arrows $S \to K$ and $S' \to K$ identify $K$ with the residue field of both $S$ and $S'$. Then there exists an unique $R$-algebra isomorphism $S \to S'$ compatible with the maps to $K$.