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Example 5.5.4.10 ($\operatorname{ \pmb {\mathcal{QC}} }$-Valued Functors). Let $\operatorname{ \pmb {\mathcal{QC}} }$ denote the $(\infty ,2)$-category of $\infty $-categories (Construction 5.4.5.1), which we view as a full simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ be a morphism of simplicial sets. Then the Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ fits into pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{\pi } & \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{ \pmb {\mathcal{QC}} }, } \]

where $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ is the $(\infty ,2)$-category of Construction 5.4.6.10 (Construction 5.4.6.8). If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ is a functor of $(\infty ,2)$-categories, then Proposition 5.4.6.9 and Remark 5.3.2.4 guarantee that $\pi : \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is an interior fibration; in particular, $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is also an $(\infty ,2)$-category.