The Stacks project

Lemma 90.25.3. Let $\mathcal{F}$ be category cofibered in groupoids over a category $\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}$. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}$ constructed from $f : U \to \mathcal{F}$ in Lemma 90.25.2. Then there is a natural morphism $[f] : [U/R] \to \mathcal{F}$ such that:

  1. $[f]: [U/R] \to \mathcal{F}$ is fully faithful.

  2. $[f]: [U/R] \to \mathcal{F}$ is an equivalence if and only if $f : U \to \mathcal{F}$ is essentially surjective.

Proof. Omitted. $\square$

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