Construction 5.3.1.5 (The Strict Transport Representation). Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories. For every object $C \in \operatorname{\mathcal{C}}$, we let $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ denote the full subcategory of $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$ spanned by those commutative diagrams
where $F$ carries each morphism of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} )$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$. The construction $C \mapsto \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ determines a functor $\operatorname{sTr}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, which we will refer to as the strict transport representation of the cocartesian fibration $U$.