The Stacks project

Proof. Let $\{ V_ a \to S\} _{a \in A}$ be the fppf covering found in Lemma 37.48.1. In other words, this covering refines $\{ S_ i \to S\}$ and each $V_ a \to S$ factors as $V_ a \to S'_ a \to S$ with $S'_ a \to S$ étale and $V_ a \to S'_ a$ surjective finite locally free.

By Remark 37.40.3 there exists a Zariski open covering $S = \bigcup U_ j$, for each $j$ a finite locally free, surjective morphism $W_ j \to U_ j$, and for each $j$ a Zariski open covering $\{ W_{j, k} \to W_ j\}$ such that the family $\{ W_{j, k} \to S\}$ refines the étale covering $\{ S'_ a \to S\}$, i.e., for each pair $j, k$ there exists an $a(j, k)$ and a factorization $W_{j, k} \to S'_ a \to S$ of the morphism $W_{j, k} \to S$. Set $T_{j, k} = W_{j, k} \times _{S'_ a} V_ a$ and everything is clear. $\square$