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Proposition 5.4.6.11.

$(1)$

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

$(2)$

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

$U_0: \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} = \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}} ) \rightarrow \operatorname{Pith}(\operatorname{ \pmb {\mathcal{QC}} }) = \operatorname{\mathcal{QC}}.$
$(3)$

The diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \ar [r] \ar [d]^-{U_0} & \operatorname{ \pmb {\mathcal{QC}} }_{\ast }^{\operatorname{lax}} \ar [d]^-{U} \\ \operatorname{\mathcal{QC}}\ar [r] & \operatorname{ \pmb {\mathcal{QC}} }}$

is a pullback square of simplicial sets.

$(4)$

The functor $U_0$ is a cocartesian fibration of $\infty$-categories.

Proof. Assertion $(1)$ follows from Proposition 5.3.5.6. Since $U$ 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. $\square$