The Stacks project

Situation 76.52.7. Let $S$ be a scheme. Let $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ be a limit of a directed system of algebraic spaces over $S$ with affine transition morphisms $g_{i'i} : Y_{i'} \to Y_ i$. We assume that $Y_ i$ is quasi-compact and quasi-separated for all $i \in I$. We denote $g_ i : Y \to Y_ i$ the projection. We fix an element $0 \in I$ and a flat morphism of finite presentation $X_0 \to Y_0$. We set $X_ i = Y_ i \times _{Y_0} X_0$ and $X = Y \times _{Y_0} X_0$ and we denote the transition morphisms $f_{i'i} : X_{i'} \to X_ i$ and $f_ i : X \to X_ i$ the projections.