The Stacks project

Remark 32.22.5. In Situation 32.22.1 Lemmas 32.22.2, 32.22.3, and 32.22.4 tell us that the category of schemes quasi-separated and of finite type over $S$ is equivalent to certain types of inverse systems of schemes over $(S_ i)_{i \in I}$, namely the ones produced by applying Lemma 32.22.3 to a diagram of the form (32.22.2.1). For example, given $X \to S$ finite type and quasi-separated if we choose two different diagrams $X \to V_1 \to S_{i_1}$ and $X \to V_2 \to S_{i_2}$ as in (32.22.2.1), then applying Lemma 32.22.4 to $\text{id}_ X$ (in two directions) we see that the corresponding limit descriptions of $X$ are canonically isomorphic (up to shrinking the directed set $I$). And so on and so forth.