Proposition 5.6.4.8 (027J)

Proposition 5.6.4.8. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, and let

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [dr]_{U} \ar [rr]^-{\theta } & & \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \ar [dl]^{U'} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]

be the commutative diagram of Remark 5.6.4.6. Then:

$(1)$: For each object $C \in \operatorname{\mathcal{C}}$, the morphism $\theta $ induces an equivalence of $\infty $-categories

\[ \theta _{C}: \{ C\} \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}). \]
$(2)$: A morphism $f$ of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is $U$-cocartesian if and only if $\theta (f)$ is a $U'$-cocartesian morphism of the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$.
$(3)$: The functor $\theta $ is an equivalence of $\infty $-categories.

Proof. Assertion $(1)$ follows from Example 5.6.4.7 and Theorem 4.6.8.9. Assertion $(2)$ follows from Example 5.6.4.3 together with the descriptions of $U$-cocartesian and $U'$-cocartesian morphisms supplied by Corollary 5.3.3.16 and Remark 5.6.2.14. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with Theorem 5.1.6.1 (since $U$ and $U'$ are cocartesian fibrations, by virtue of Corollary 5.3.3.16 and Proposition 5.6.2.2). $\square$