The Stacks project

Proof. Choose an étale covering $\{ Y_ i \to Y\}$ where $Y_ i$ is an affine scheme. For each $i$ and $j$ choose a affine open covering $Y_ i \times _ Y Y_ j = \bigcup Y_{ijk}$. Set $\mathcal{X}_ i = Y_ i \times _ Y \mathcal{X}$ and $\mathcal{X}_{ijk} = Y_{ijk} \times _ Y \mathcal{X}$. Let $g : \mathcal{X} \to W$ be a morphism towards an algebraic space. Then we consider the diagram

The assumption that $\mathcal{X}_ i \to Y_ i$ is a categorical moduli space, produces a unique dotted arrow $h_ i : Y_ i \to W$. The assumption that $\mathcal{X}_{ijk} \to Y_{ijk}$ is a categorical moduli space, implies the restriction of $h_ i$ and $h_ j$ to $Y_{ijk}$ are equal. Hence $h_ i$ and $h_ j$ agree on $Y_ i \times _ Y Y_ j$. Since $Y = \coprod Y_ i / \coprod Y_ i \times _ Y Y_ j$ (by Spaces, Section 65.9) we conclude that there is a unique morphism $Y \to W$ through which $g$ factors. Thus $f$ is a categorical moduli space. The same argument applies after a flat base change, hence $f$ is a uniform categorical moduli space. $\square$