I have some trouble with the last step of the proof of the following theorem in Lei Fu's Etale Cohomology.
The statement of the theorem is:
Proposition 2.3.12: Let $S$ be a scheme, $S_0$ a closed subscheme of $S$ with the same underlying topological space. Then the functor $X\to X\times_SS_0$ from the category of etale $S$-schemes to etale $S_0$-schemes is an equivalence of categories.
In the proof he first shows that the functor is fully faithful. Then for any etale $S_0$ scheme $X_0$, he proves that we can find an open covering $\{U_{\alpha_0}\}$ and etale $S$-schemes $U_\alpha$ such that $U_\alpha\times_SS_0=U_{\alpha_0}$. My question is just how can we glue these $U_{\alpha_0}$ together to get the desired $S$ scheme $U$?
you have to use fully faithfulness. this is a standard technique when you want to construct an object with some property $P$ over each member of a site you first prove uniqueness (in this case this implied by fully faithfulness) then uniqueness guarantee that you can glue these objects(because there is only one object with property $P$ over the intersection) hence it is enough to construct your object locally.