Let $X \overset{f^\prime}\longrightarrow S^\prime \overset\pi\longrightarrow S$ be morphisms of schemes such that $f^\prime$ is proper with geometrically connected fibers and $\pi$ is integral. Set $f = \pi \circ f^\prime$. Is it true that $S^\prime$ is the normalization of $S$ in $X$.



Uniqueness of Stein factorization
No. Take $S=S’$ a cuspidal curve, $\pi$ the identity, and $f’=f: X\to S’=S$ the normalization.


