I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu.
$\mathrm{Proposition} 1.6.6$ Let $g:S'\rightarrow S$ be a quasi-compact faithfully flat morphism, and let $\mathcal F$ be a quasi-coherent $\mathcal O_S$-module. Then $g^* \mathcal F$ is locally of finite type(resp. locally of finite presentation,resp. locally free of finite rank) if and only if $\mathcal F$ is so.
The author reduces the proof to the case where both S' and S are affine, but the detail is left to the reader.
How can I confirm this reduction? S' and S seem to be schemes but not necessarily Noetherian.