The Stacks project

Situation 32.8.1. Let $S = \mathop{\mathrm{lim}}\nolimits S_ i$ be a limit of a directed system of schemes with affine transition morphisms (Lemma 32.2.2). Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_ i : X_ i \to Y_ i$ be the base change of $f_0$ to $S_ i$ and let $f : X \to Y$ be the base change of $f_0$ to $S$.