Example 5.6.5.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\operatorname{\mathcal{E}}^{0} \subseteq \operatorname{\mathcal{E}}$ be the simplicial subset consisting of those simplices $\Delta ^ n \rightarrow \operatorname{\mathcal{E}}$ which carry each edge of $\Delta ^ n$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$, so that $U$ restricts to a left fibration $U^{0}: \operatorname{\mathcal{E}}^{0} \rightarrow \operatorname{\mathcal{C}}$ (see Corollary 5.1.4.16). If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for $U$, then $\mathscr {F}^{\simeq }: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a covariant transport representation for $U^{0}$, where $\mathscr {F}^{\simeq }$ denotes the composition of $\mathscr {F}$ with the functor
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
\[ \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{S}}\quad \quad \operatorname{\mathcal{D}}\mapsto \operatorname{\mathcal{D}}^{\simeq } \]