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Construction 7.5.2.1 (Homotopy Limits of $\infty $-Categories). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories, let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the weighted nerve of $\mathscr {F}$ (Definition 5.3.3.1), and let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ the cocartesian fibration Proposition 5.3.3.15. We let $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ denote the full subcategory

\[ \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \subseteq \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \]

whose objects are functors $G: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ which satisfy $U \circ G = \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ and which carry each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Notation 5.3.1.10). We will refer to $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ as the homotopy limit of the diagram $\mathscr {F}$.