I am asking this question in order to understand the answer to this five years old question. I also added a comment there, but the user answering the question hasn't been active for years so I don't expect to get an answer there.
In the question mentioned above, it is claimed that a smooth morphism $X\to Y$ has smooth source, i.e. that $X$ is smooth. I was unable to find this statement anywhere and I have no idea at all about how to see this is true.
Any help is kindly appreciated.
If $Y$ is smooth and $f:X \to Y$ is smooth then $X$ is smooth (this is Hartshorne III.10.1(c) for $Z$ a point). If $Y$ is singular then $X$ is singular as well (the simplest example is when $f$ is an isomorphism).