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Remark 5.2.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a functor $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i \} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$. Suppose that we are given a sequence of functors

\[ \operatorname{\mathcal{C}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{C}}(1) \xrightarrow { F(2)} \operatorname{\mathcal{C}}(2) \xrightarrow { F(3)} \cdots \xrightarrow {F(n)} \operatorname{\mathcal{C}}(n) \]

and a morphism of simplicial sets $U: M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}$ satisfying condition $(1)$ of Definition 5.2.6.12. Since the collection of $\pi $-cocartesian morphisms of $\operatorname{\mathcal{C}}$ is closed under composition (Corollary 5.1.2.4), we can replace $(2)$ with the following a priori weaker condition:

$(2')$

For every integer $1 \leq i \leq n$ and every object $C \in \operatorname{\mathcal{C}}(i-1)$, the composition

\[ \Delta ^1 \times \{ C\} \rightarrow \operatorname{N}_{\bullet }( \{ i - 1 < i \} ) \times \operatorname{\mathcal{C}}(i-1) \rightarrow M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

is a $\pi $-cocartesian edge of $\operatorname{\mathcal{C}}$.

If $\pi $ is a cocartesian fibration, then a morphism of $\operatorname{\mathcal{C}}$ is $\pi $-cocartesian if and only if it is locally $\pi $-cocartesian (Remark 5.1.4.5). In this case, we can restate $(2')$ as follows:

$(2'')$

For every integer $1 \leq i \leq n$, the composition

\[ \operatorname{N}_{\bullet }( \{ i-1 < i \} ) \times \operatorname{\mathcal{C}}(i-1) \rightarrow M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

witnesses the functor $F(i): \operatorname{\mathcal{C}}(i-1) \rightarrow \operatorname{\mathcal{C}}(i)$ as given by covariant transport along the edge $\operatorname{N}_{\bullet }( \{ i-1 < i \} ) \subseteq \Delta ^ n$ (in the sense of Definition 5.2.2.1).