The Stacks project

Proof. For $n = 1$ this is Limits of Spaces, Lemma 70.19.1. For $n > 1$ the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that $g_ i : Y_ i \to U_ i$ are objects of $\mathcal{C}_{U_ i, u_ i}$. Then by the case $n = 1$ we can find $f'_ i : Y'_ i \to X$ which are isomorphisms over $X \setminus \{ x_ i\} $ and whose base change to $U_ i$ is $f_ i$. Then we can set

This is an object of $\mathcal{C}_{X, \{ x_1, \ldots , x_ n\} }$ whose base change by $U_ i \to X$ recovers $g_ i$. Thus the functor is essentially surjective. We omit the proof of fully faithfulness. $\square$