# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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 4.6.7.9. Assertion $(2)$ follows from Example 5.6.4.3 together with the descriptions of $U$-cocartesian and $U'$-cocartesian morphisms supplied by Proposition 5.3.3.15 and Remark 5.6.2.14. 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 5.3.3.15 and 5.6.2.2). $\square$