$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 5.3.3.15. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Assume that:
For each object $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a cocartesian fibration of simplicial sets.
For each morphism $u: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the morphism $\mathscr {F}(u): \mathscr {F}(C) \rightarrow \mathscr {F}(D)$ carries $\alpha _{C}$-cocartesian edges of $\mathscr {F}(C)$ to $\alpha _ D$-cocartesian edges of $\mathscr {F}(D)$.
Then:
- $(1)$
The induced map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets.
- $(2)$
Let $(f,e): (C,x) \rightarrow (D,y)$ be an edge of the simplicial set $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Remark 5.3.3.4). Then $(f,e)$ is $U$-cocartesian if and only if $e: \mathscr {F}(f)(x) \rightarrow y$ is an $\alpha _{D}$-cocartesian edge of the simplicial set $\mathscr {F}(D)$.
Proof.
By virtue of Proposition 5.1.4.8 and Remark 5.3.3.7, we may assume without loss of generality that $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ for some nonnegative integer $n$. We proceed by induction on $n$. If $n = 0$, then $U$ can be identified with the cocartesian fibration $\alpha _0: \mathscr {F}(0) \rightarrow \mathscr {G}(0)$ (Example 5.3.3.2), so that assertions $(1)$ and $(2)$ are immediate. Let us therefore assume that $n > 0$, so that $\operatorname{\mathcal{C}}$ can be identified with the cone $\operatorname{\mathcal{C}}_{0}^{\triangleright }$ for $\operatorname{\mathcal{C}}_0 = [n-1]$. Set $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ and $\mathscr {G}_0 = \mathscr {G}|_{ \operatorname{\mathcal{C}}_0 }$. It follows from our inductive hypothesis that $U$ restricts to a cocartesian fibration $U_0: \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_0}(\operatorname{\mathcal{C}}_0)$ is a cocartesian fibration of $\infty $-categories, and that an edge of $\operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0)$ is $U_0$-cocartesian if and only if it satisfies the criterion described in $(2)$. It follows that the functor $\operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \mathscr {F}(n)$ described in Example 5.3.3.12 carries $U_0$-cocartesian morphisms to $\alpha _{n}$-cocartesian morphisms of the $\infty $-category $\mathscr {F}(n)$. Unwinding the definitions, we can identify $U$ with the map of relative joins
\[ \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \star _{\mathscr {F}(n)} \mathscr {F}(n) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_0}(\operatorname{\mathcal{C}}_0) \star _{\mathscr {G}(n)} \mathscr {G}(n). \]
Assertions $(1)$ and $(2)$ now follow from Lemma 5.2.3.17.
$\square$