diff --git a/morphisms.tex b/morphisms.tex
index 0e1a7cba..7c828b94 100644
--- a/morphisms.tex
+++ b/morphisms.tex
@@ -14214,7 +14214,8 @@ \section{Normalization}
 $X'$ satisfies the assumptions under which the normalization
 is defined. Let $f' : Y' \to X'$ be the morphism
 (\ref{equation-generic-points}) constructed starting with $X'$.
-As $\alpha$ is birational it is clear that $Y' = Y$ and $f = \alpha \circ f'$.
+As $\alpha$ is locally birational it is clear that
+$Y' = Y$ and $f = \alpha \circ f'$.
 Hence the factorization $X^\nu \to X' \to X$ exists
 and $X^\nu \to X'$ is the normalization of $X'$ by
 Lemma \ref{lemma-characterize-normalization}. This proves (3).
@@ -14262,7 +14263,7 @@ \section{Normalization}
 Then the lemma follows either from the local description in
 Lemma \ref{lemma-description-normalization}
 or from Lemma \ref{lemma-normalization-normal} part (3) because
-$\coprod Z_i \to X$ is integral and birational (as $X$ is reduced
+$\coprod Z_i \to X$ is integral and locally birational (as $X$ is reduced
 and has locally finitely many irreducible components).
 \end{proof}