Let $S$ be a scheme and $X\to Y$ be a morphism over $S$. Then we have an induced homomorphism of sheaves $h\_X=\operatorname{Hom}\_S({-}, X)\to h\_Y=\operatorname{Hom}\_S({-}, Y)$ over the small étale site $S\_\text{étale}$.


Question: When is $h\_X\to h\_Y$ a surjective homomorphism between sheaves of sets over $S\_\text{étale}$?


I find this result in the [comment](https://math.stackexchange.com/questions/22008/%C3%89tale-local-sections-of-a-smooth-surjective-morphism#comment633866_22008) of Ehsan M. Kermani under [Étale Local Sections of a Smooth Surjective Morphism](https://math.stackexchange.com/questions/22008/%C3%89tale-local-sections-of-a-smooth-surjective-morphism). Is there any proof or reference of this claim?



Surjective sheaf homomorphisms induced by morphisms of schemes
**Proposition.** Let $f\colon X\to Y$ be a surjective morphism such that $\forall y\in Y$ there exists a point $x\in f^{-1}(y)$ where $f$ is smooth (i.e. every fiber has a non-reduced point). Then $h\_X\to h\_Y$ is surjective on the (small or big) étale site.


**Proof:** Take $f\in h\_Y(S)$. The property of having a smooth point in every fiber is stable under base change so we may assume $S=Y$. Consider the open subset $X^\circ$ of $X$ where $f$ is smooth. The restriction $f^\circ$ of $f$ to $X^\circ$ is smooth and surjective. Apply EGA IV 4 Cor. 17.16.3 (ii) to $f^\circ$. We find an étale covering $Y'\to Y$ such that $Y'\times\_Y X$ admits a section $s\colon Y'\to Y'\times\_Y X$ which is what we need.


**Remark 1.** By EGA IV 4 Cor. 17.16.2 if $f\colon X\to Y$ is locally of finite presentation and faithfully flat then it admits a section after a faithfully flat base change. Therefore, in this case $h\_X$ surjects over $h\_Y$ on the (small or big) fppf site.


**Remark 2.** The property in the proposition is equivalent to the surjectivity of $h\_X\to h\_Y$. Indeed, the surjectivity implies that $f$ admits a section étale locally on $Y$.


