The Stacks project

Definition 89.21.9. Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$.

  1. The assignment $T \mapsto (U(T), R(T), s, t, c)$ determines a functor $\mathcal{C} \to \textit{Groupoids}$. The quotient category cofibered in groupoids $[U/R] \to \mathcal{C}$ is the category cofibered in groupoids over $\mathcal{C}$ associated to this functor (as in Remarks 89.5.2 (9)).

  2. The quotient morphism $U \to [U/R]$ is the morphism of categories cofibered in groupoids over $\mathcal{C}$ defined by the rules

    1. $x \in U(T)$ maps to the object $(T, x) \in \mathop{\mathrm{Ob}}\nolimits ([U/R](T))$, and

    2. $x \in U(T)$ and $f : T \to T'$ give rise to the morphism $(f, \text{id}_{U(f)(x)}): (T, x) \to (T, U(f)(x))$ lying over $f : T \to T'$.

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