Remark 7.4.2.10 (Composition). Let $\operatorname{\mathcal{C}}$ be a simplicial set. Let $\mathscr {F}, \mathscr {F}', \mathscr {F}'': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be covariant transport representations for left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$, and $U'': \operatorname{\mathcal{E}}'' \rightarrow \operatorname{\mathcal{C}}$. Suppose we are given a commutative diagram
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$. If $\Gamma : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ is compatible with $\gamma $ and $\Gamma ': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}''$ is compatible with $\gamma '$, then the composite map $(\Gamma ' \circ \Gamma ): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}''$ is compatible with $\gamma ''$.