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Proposition 5.4.6.11.

$(1)$

The simplicial set $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ is an $\infty $-category.

$(2)$

The projection map $\widetilde{V}: \operatorname{ \pmb {\mathcal{QC}} }_{\ast } = \operatorname{ \pmb {\mathcal{QC}} }_{\Delta ^{0} / } \rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ restricts to a functor

\[ V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} = \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}} ) \rightarrow \operatorname{Pith}(\operatorname{ \pmb {\mathcal{QC}} }) = \operatorname{\mathcal{QC}}. \]
$(3)$

The diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [r] \ar [d]^-{V} & \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}} \ar [d]^-{ \widetilde{V} } \\ \operatorname{\mathcal{QC}}\ar [r] & \operatorname{ \pmb {\mathcal{QC}} }} \]

is a pullback square of simplicial sets.

$(4)$

The functor $V$ is a cocartesian fibration of $\infty $-categories.

Proof. Assertion $(1)$ follows from Proposition 5.3.5.6. Since $\widetilde{V}$ is an interior fibration (Proposition 5.4.6.9), assertions $(2)$ and $(3)$ follow from Proposition 5.3.7.9. Assertion $(4)$ is a special case of Corollary 5.3.7.10. $\square$