TITLE: products and smooth/étale/unramified morphisms
QUESTION [6 upvotes]: Let $X$, $Y$ and $Z$ be Noetherian schemes. 
If $f: Y \to X$ and $g: Z \to X$ are morphisms of finite type, such that at each point of $X$, at least one of the two morphisms is smooth/étale/unramified (at all points of its inverse image), can we conclude that the induced morphism $Y \times_X Z \to X$ is smooth/étale/unramified everywhere? 
If not, which results can we obtain?
(In his textbook on Algebraic Geometry, Liu asks to prove that the answer is always "yes"...)
EDIT. So, indeed, the problem statement in the book is wrong...

REPLY [5 votes]: Smooth, unramified, and etale morphisms are stable under base change, so $Y\times_X Z \to Z$ is {smooth, unramified, etale} if $Y\to X$ is {smooth, unramified, etale}.