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Proposition 5.5.6.8. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, and let

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

be the commutative diagram of Remark 5.5.6.6. Then:

$(1)$

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

\[ \theta _{C}: \{ C\} \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}). \]
$(2)$

A morphism $f$ of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ 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 5.5.6.7 and Theorem 4.6.6.9. Assertion $(2)$ follows from Example 5.5.6.3 together with the descriptions of $U$-cocartesian and $U'$-cocartesian morphisms supplied by Proposition 5.5.3.13 and Remark 5.5.4.14. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with Theorem 5.1.5.1 (since $U$ and $U'$ are cocartesian fibrations, by virtue of Propositions 5.5.3.13 and 5.5.4.2. $\square$