Example 5.6.5.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $K$ be another simplicial set, and form a pullback diagram
\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [d]^{U'} & \operatorname{Fun}(K, \operatorname{\mathcal{E}}) \ar [d]^{U \circ } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Fun}(K, \operatorname{\mathcal{C}}). } \]
If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for $U$, then $\mathscr {F}^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for $U'$, where $\mathscr {F}^{K}$ is obtained by composing $\mathscr {F}$ with the functor $\operatorname{Fun}(K, \bullet ): \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$. See Example 5.6.2.20.