Lemma 58.10.7. Let $X$ be an irreducible, geometrically unibranch scheme. For any nonempty open $U \subset X$ the canonical map
is surjective. The map (58.10.6.1) $\pi _1(\eta , \overline{\eta }) \to \pi _1(X, \overline{\eta })$ is surjective as well.
Proof. By Lemma 58.8.3 we may replace $X$ by its reduction. Thus we may assume that $X$ is an integral scheme. By Lemma 58.4.1 the assertion of the lemma translates into the statement that the functors $\textit{FÉt}_ X \to \textit{FÉt}_ U$ and $\textit{FÉt}_ X \to \textit{FÉt}_\eta $ are fully faithful.
The result for $\textit{FÉt}_ X \to \textit{FÉt}_ U$ follows from Lemma 58.10.5 and the fact that for a local ring $A$ which is geometrically unibranch its strict henselization has an irreducible spectrum. See More on Algebra, Lemma 15.106.5.
Observe that the residue field $\kappa (\eta ) = \mathcal{O}_{X, \eta }$ is the filtered colimit of $\mathcal{O}_ X(U)$ over $U \subset X$ nonempty open affine. Hence $\textit{FÉt}_\eta $ is the colimit of the categories $\textit{FÉt}_ U$ over such $U$, see Limits, Lemmas 32.10.1, 32.8.3, and 32.8.10. A formal argument then shows that fully faithfulness for $\textit{FÉt}_ X \to \textit{FÉt}_\eta $ follows from the fully faithfulness of the functors $\textit{FÉt}_ X \to \textit{FÉt}_ U$. $\square$
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