# Surjective sheaf homomorphisms induced by morphisms of schemes

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 of Ehsan M. Kermani under Étale Local Sections of a Smooth Surjective Morphism. Is there any proof or reference of this claim?

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 $$Y$$.