The Stacks project

Proof. (Recall that pullbacks of descent data are defined in Descent, Definition 35.34.7.) Set $X_ i = X \times _ S S_ i$. Denote $(V_ i, \varphi _ i)$ the pullback of $(V, \varphi )$ to $X_ i/S_ i$. By assumption (1) we can find an étale morphism $U_ i \to S_ i$ which comes with an isomorphism $X_ i \times _{S_ i} U_ i \to V_ i$ compatible with $can$ and $\varphi _ i$. By assumption (2) we obtain isomorphisms $\psi _{ij} : U_ i \times _{S_ i} (S_ i \cap S_ j) \to U_ j \times _{S_ j} (S_ i \cap S_ j)$. By assumption (3) these isomorphisms satisfy the cocycle condition so that $(U_ i, \psi _{ij})$ is a descend datum for the Zariski covering $\{ S_ i \to S\} $. Then Descent, Lemma 35.35.10 (which is essentially just a reformulation of Schemes, Section 26.14) tells us that there exists a morphism of schemes $U \to S$ and isomorphisms $U \times _ S S_ i \to U_ i$ compatible with $\psi _{ij}$. The isomorphisms $U \times _ S S_ i \to U_ i$ determine corresponding isomorphisms $X_ i \times _ S U \to V_ i$ which glue to a morphism $X \times _ S U \to V$ compatible with the canonical descent datum and $\varphi $. $\square$