Definition 83.4.4. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces over $B$ closed under fibre products. Let $j = (t, s) : R \to U \times _ B U$ be pre-relation in $\mathcal{C}$, and let $U \to X$ be an $R$-invariant morphism with $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.
We say $U \to X$ is a universal categorical quotient in $\mathcal{C}$ if for every morphism $X' \to X$ in $\mathcal{C}$ the morphism $U' = X' \times _ X U \to X'$ is the categorical quotient in $\mathcal{C}$ of the base change $j' : R' \to U'$ of $j$.
We say $U \to X$ is a uniform categorical quotient in $\mathcal{C}$ if for every flat morphism $X' \to X$ in $\mathcal{C}$ the morphism $U' = X' \times _ X U \to X'$ is the categorical quotient in $\mathcal{C}$ of the base change $j' : R' \to U'$ of $j$.
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