# On the flatness of certain morphism

Assume $X, Z, S$ are shemes of finite type over $\mathbb{C}$, $X$ is also irreducible and reduced, $\phi: Z\to S$ is affine flat morphism with reduced connected fibers, $\psi: Z\to X$ is such that $\phi \times \psi$ is finite. Is $\text{Im}(\phi \times \psi)\to S$ flat? Maybe, under some additional conditions?

For example, let $S=\mathbb{A}^1$, $Z=X=\mathbb{A}^2$, $\phi$ is given by $(x,y)\mapsto x$, $\psi$ is given by $(x,y)\mapsto (xy, y^2)$. The morphism $\phi\times\psi$ is bijective outside the fiber $Z_0$, where it is 2 to 1. However, $\text{Im}(\phi \times \psi)\to S$ is flat.

Thanks.

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