In Olsson's Algebraic spaces and stacks there is the following lemma:
Lemma 1.1.6 Let $f: X \to Y$ be a flat morphism of locally noetherian schemes that is locally of finite type (or more generally, $f$ is a fppf morphism without $X$ and $Y$ being locally noetherian), and let $Y = \bigcup_i U_i$ be an open affine cover. Then for each $i$ there is a Zariski covering $f^{-1}(U_i) = \bigcup_j V_{ij}$ with $V_{ij}$ quasi-compact and $f(V_{ij}) = U_i$.
When this is applied, Olsson writes that the restricted morphism $f_{ij} = f|_{V_{ij}}: V_{ij} \to U_i$ is now quasi-compact and quasi-separated. Why is that the case? Any help would be appreciated!