This is related to this old question, which doesn't seem to address my exact situation.
Suppose I have a smooth morphism $\varphi: X\to Y$ of smooth varieties over an algebraically closed field, and let $E\subset X$ and $F\subset Y$ be such that
1) $F\subset Y$ is an irreducible, smooth closed subscheme,
2) $E\subset X$ is irreducible, reduced closed subscheme, and
3) the restriction $\varphi|_{E}:E\to F$ is surjective with smooth equidimensional (though not necessarily connected) fibers.
My question is: is this map flat? smooth? or are there simply counterexamples to either? The above cited question did not assume the equidimensionality of the fibers (hence, the counterexample of a blow up), nor the smoothness of the base.
The answers to this other related question do not help me either, since here I assume that $F$ and the fibers of the morphism are smooth.
If I knew that $E$ were Cohen-Macaulay, then the restriction would be flat by miracle flatness, so I might hope that this is forced by the restrictions, but am not sure how to prove it.