Definition 89.5.1. Let $\mathcal{C}$ be a category. A category cofibered in groupoids over $\mathcal{C}$ is a category $\mathcal{F}$ equipped with a functor $p: \mathcal{F} \to \mathcal{C}$ such that $\mathcal{F}^{opp}$ is a category fibered in groupoids over $\mathcal{C}^{opp}$ via $p^{opp}: \mathcal{F}^{opp} \to \mathcal{C}^{opp}$.

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