Example 11.9.1.4. In the situation of Theorem 11.9.1.1, suppose that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex, so that $\operatorname{Tw}(\operatorname{\mathcal{C}})$ can be identified with the nerve of the partially ordered set $Q = \{ (i,j) \in [n]^{\operatorname{op}} \times [n]: i \leq j \} $. For $0 \leq i \leq n$, let $\operatorname{\mathcal{E}}_{i}$ denote the fiber $\{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{E}}$. Then an object of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ can be identified with a functor $F: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{\mathcal{E}}$ having the property that $F(i,j) \in \operatorname{\mathcal{E}}_{j}$ for $0 \leq i \leq j \leq n$. In this case, conditions $(1)$ and $(2)$ of Theorem 11.9.1.1 can be stated more concretely as follows:
- $(1)$
For $0 \leq i \leq n$, the image $F(i,i)$ is an initial object of the $\infty $-category $\operatorname{\mathcal{E}}_{i}$.
- $(2)$
For $0 \leq i \leq j \leq k \leq n$, the functor $F$ determines a $U$-cocartesian morphism $F(i,j) \rightarrow F(i,k)$ in the $\infty $-category $\operatorname{\mathcal{E}}$.
Moreover, since the collection of $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ contains all identity morphisms (Proposition 5.1.1.8) and is closed under composition (Corollary 5.1.2.4), it suffices to verify condition $(2)$ under the additional assumption that $j = k-1$.