Let $S$ be a scheme. It is known that the smooth topology on $\textrm{Sch}_S$ is equivalent to the étale topology, basically because every smooth covering can be refined to an étale covering.
Question. Is it true, then, that all properties P of morphisms of schemes that are of local nature for the étale topology are also of local nature for the smooth topology?
I believe this would simplify a lot some arguments, which is also why I believe the answer is no... Let me give you one example.
Let P be a property of morphisms of schemes, assumed to be local on the source and target for the étale topology.
Definition. A map of Deligne-Mumford $f:\mathcal X\to \mathcal Y$ has P if, given a commutative diagram
$$\require{AMScd} \begin{CD} X @>{x}>> \mathcal X\\ @V{g}VV @VV{f}V \\ Y @>{y}>> \mathcal Y \end{CD}$$ where $X,Y$ are schemes and $x,y$ are $\color{red}{\textrm{étale}}$ and surjective, the map $g$ has P.
My question above is equivalent to the following:
Question'. Can we replace $\color{red}{\textrm{étale}}$ by $\color{red}{\textrm{smooth}}$ in the above definition?
