The Stacks project

Lemma 105.12.3. With assumption and notation as in Lemma 105.12.2. Then $f$ is a (uniform) categorical moduli space if and only if $\phi $ is a (uniform) categorical quotient. Similarly for moduli spaces in a full subcategory.

Proof. It is immediate from the $1$-to-$1$ correspondence established in Lemma 105.12.2 that $f$ is a categorical moduli space if and only if $\phi $ is a categorical quotient (Quotients of Groupoids, Definition 82.4.1). If $Y' \to Y$ is a morphism, then $U' = Y' \times _ Y U \to Y' \times _ Y \mathcal{X} = \mathcal{X}'$ is a surjective, flat, locally finitely presented morphism as a base change of $U \to \mathcal{X}$ (Criteria for Representability, Lemma 96.17.1). And $R' = Y' \times _ Y R$ is equal to $U' \times _{\mathcal{X}'} U'$ by transitivity of fibre products. Hence $\mathcal{X}' = [U'/R']$, see Algebraic Stacks, Remark 93.16.3. Thus the base change of our situation to $Y'$ is another situation as in the statement of the lemma. From this it immediately follows that $f$ is a uniform categorical moduli space if and only if $\phi $ is a uniform categorical quotient. $\square$

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