Suppose X, Y, Z are k-varieties and $f: X to Z$ factors through $f': X to Y$ and $g: Y to Z$. Suppose all of f, f', g are surjective. Assume that for $z in Z$, the fibre $f^{-1} (z)$ is reduced. Then, is the fibre $g^{-1} (z)$ always reduced?
If not, when will it be true?
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