The Stacks project

Proof. For $n = 1$ this is Limits, Lemma 32.21.1. For $n > 1$ the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that $g_ i : Y_ i \to S_ i$ are objects of $FP_{S_ i, s_ i}$. Then by the case $n = 1$ we can find $f'_ i : X'_ i \to S$ of finite presentation which are isomorphisms over $S \setminus \{ s_ i\} $ and whose base change to $S_ i$ is $g_ i$. Then we can set

This is an object of $FP_{S, \{ s_1, \ldots , s_ n\} }$ whose base change by $S_ i \to S$ recovers $g_ i$. Thus the functor is essentially surjective. We omit the proof of fully faithfulness. $\square$