Kerodon

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Remark 5.5.3.15. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and let $\pi : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Proposition 5.5.3.13. Then the homotopy transport representation $\operatorname{hTr}_{\pi }: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 5.2.3.2 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 induced map $\mathscr {F}(f): \mathscr {F}(C) \rightarrow \mathscr {F}(D)$ agrees with the functor

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

given by covariant transport along $f$, which follows immediately from Proposition 5.2.4.16 (together with Example 5.5.3.8).