# normalization of a bijection

Let $X^\nu,Y^\nu$ be normalizations of affine varieties $X$ and $Y.$ If a morphism $f:X\to Y$ is a bijection, does it imply that its lift $f^\nu: X^\nu\to Y^\nu$ is an isomorphism?

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Let $Y$ be an affine curve which is smooth away from a single node. We obtain $X$ from the normalization of $Y$ by removing one of the points mapping to the node of $Y$. The map from $X$ to $Y$ is a bijection, but the map $X^\nu\to Y^\nu$ is not an isomorphism.
I guess the right assumption that would make things work is that $f$ is proper and separable. –  Piotr Achinger Oct 28 '12 at 21:41