Kerodon

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Proposition 5.5.6.9. The simplicial set $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ is an $(\infty ,2)$-category. Moreover, the projection map $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}} \rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ is an interior fibration of $(\infty ,2)$-categories.

Proof. It follows from Proposition 5.5.5.2 that $\operatorname{ \pmb {\mathcal{QC}} }$ is an $(\infty ,2)$-category. The desired conclusion now follows from Corollary 5.4.3.4 and Proposition 5.4.3.1. $\square$