Lemma 29.53.7. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X'$ be the normalization of $X$ in $Y$. Then the normalization of $X'$ in $Y$ is $X'$.

Proof. If $Y \to X'' \to X'$ is the normalization of $X'$ in $Y$, then we can apply Lemma 29.53.4 to the composition $X'' \to X$ to get a canonical morphism $h : X' \to X''$ over $X$. We omit the verification that the morphisms $h$ and $X'' \to X'$ are mutually inverse (using uniqueness of the factorization in the lemma). $\square$

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