Let $f:X\to S$ be a morphism of schemes. Let $x\in X$ and let $\varphi:(U,u)\to (X,x)$ be an etale neighborhood of $x$ over $S$. Let $g = f\varphi:U\to S$ denote the structure morphism.
Assume that $g$ is smooth at $u$. Does it follow that $f$ is smooth at $x = \varphi(u)$?
In other words, to check smoothness of a morphism at a point, is it enough to check it at an etale neighborhood of the said point?
It boils down to the following algebraic statement: if $R,A,B$ are rings with morphisms $R\xrightarrow{f} A \xrightarrow{\varphi} B$, denoting by $g = \varphi f:R\to B$ the composition, is it true that $\varphi$ étale and $g$ smooth implies $f$ smooth?
This is true: it is essentially Lemma 036U of Stacks Project (https://stacks.math.columbia.edu/tag/036T).