Exercise 11.5.0.38. Suppose we are given a finite sequence of $\infty $-categories $\{ \operatorname{\mathcal{E}}(m) \} _{0 \leq im \leq n}$ and functors
\[ \operatorname{\mathcal{E}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{E}}(1) \xrightarrow { F(2) } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {F(n)} \operatorname{\mathcal{E}}(n). \]
Let $\operatorname{\mathcal{E}}$ denote the iterated relative join
\[ ((( \operatorname{\mathcal{E}}(0) \star _{\operatorname{\mathcal{E}}(1)} \operatorname{\mathcal{E}}(1) ) \star _{\operatorname{\mathcal{E}}(2)} \operatorname{\mathcal{E}}(2)) \star \cdots ) \star _{\operatorname{\mathcal{E}}(n)} \operatorname{\mathcal{E}}(n). \]
Show that the associated projection map $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$ is a cocartesian fibration whose homotopy transport representation $\operatorname{hTr}_{U}: [n] \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ is the diagram
\[ \operatorname{\mathcal{E}}(0) \xrightarrow { [F(1)] } \operatorname{\mathcal{E}}(1) \xrightarrow { [F(2)] } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {[F(n)]} \operatorname{\mathcal{E}}(n). \]
For a more general statement, see Proposition 11.6.0.19 and Remark 5.3.3.22.