Notation 5.6.8.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. We let $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ denote the simplicial subset of the fiber product
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \]
whose $n$-simplices are diagrams
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^ n \times \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{\mathscr {F}} } \ar [d]^{\operatorname{id}_{\Delta ^{n}} \times U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \Delta ^{n} \times \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F} } & \operatorname{\mathcal{QC}}} \]
which witness $\mathscr {F}$ as a covariant transport representation for the cocartesian fibration $(\operatorname{id}_{ \Delta ^{n} } \times U): \Delta ^{n} \times \operatorname{\mathcal{E}}\rightarrow \Delta ^{n} \times \operatorname{\mathcal{C}}$.