Kerodon

Example 7.4.2.12. In the situation of Proposition 7.4.2.11, suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ be the nerve of a category $\operatorname{\mathcal{C}}_0$, and that $\mathscr {F}$ and $\mathscr {F}'$ are given as the (homotopy coherent) nerves of the strict transport representations $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}_0}, \operatorname{sTr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}$. In this case, we can take $\gamma : \mathscr {F} \rightarrow \mathscr {F}'$ to be the (homotopy coherent) nerve of the natural transformation $\gamma _0: \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}_0} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}_0}$ given by composition with $\Gamma $. See Examples 7.4.2.8 and 7.4.1.12.