Assume that

a Cartesian square of Noetherian schemes such that the vertical arrows are proper and surjective and the horizontal morphisms are regular. If $U$, $V$ are $V^{\prime}$ are irreducible, $V$ and $V^{\prime}$ are normal and $g_{*}\mathcal{O}_{V^{\prime}}=\mathcal{O}_{V}$ then $U^{\prime}$ is normal and irreducible.