Remark 5.3.3.22 (026C): The Homotopy Transport Representation

Remark 5.3.3.22 (The Homotopy Transport Representation). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Corollary 5.3.3.16. Then the homotopy transport representation

\[ \operatorname{hTr}_{\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \]

of Construction 5.2.5.2 is canonically isomorphic to the composition $\operatorname{\mathcal{C}}\xrightarrow {\mathscr {F}} \operatorname{QCat}\twoheadrightarrow \mathrm{h} \mathit{\operatorname{QCat}}$. To prove this, it suffices to observe that for every morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, the functor

\[ \mathscr {F}(f): \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \{ D\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq \mathscr {F}(D) \]

is given by covariant transport along $f$, which follows immediately from Proposition 5.2.3.15 and Example 5.3.3.14.