Let $S$ be an integral Dedekind scheme.

Let $f:X\longrightarrow \mathbf{P}^1_{S}$ be a finite flat surjective morphism, where $X$ is an integral normal scheme.

Let $\eta$ be the generic point of $S$. Note that $f_\eta:X_\eta\longrightarrow \mathbf{P}^1_{K(S)}$ is a finite morphism of curves over $K(S)$.

**Question.** Is $f$ the normalization of $\mathbf{P}^1_S$ in the function field of $X_\eta$?