Definition 7.4.2.2 (05R7)

Definition 7.4.2.2. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{E}}\ar [rr]^{\Gamma } \ar [dr]^{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}, & } \]

where $U$ and $U'$ are left fibrations having covariant transport representations $\mathscr {F} = \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ and $\mathscr {F}' = \operatorname{Tr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}}$, respectively. We say that a natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {F}'$ is compatible with $\Gamma $ if the diagram

\[ \xymatrix { & \underline{\Delta ^0}_{\operatorname{\mathcal{E}}} \ar [dl]^{ \alpha } \ar [dr]_{ \alpha '|_{ \operatorname{\mathcal{E}}}} & \\ \mathscr {F}|_{ \operatorname{\mathcal{E}}} \ar [rr]^{ \gamma |_{ \operatorname{\mathcal{E}}} } & & \mathscr {F}'|_{ \operatorname{\mathcal{E}}} } \]

commutes up to homotopy (in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}})$). Here $\alpha : \underline{\Delta ^0}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}|_{\operatorname{\mathcal{E}}}$ and $\alpha ': \underline{\Delta ^0}_{\operatorname{\mathcal{E}}' } \rightarrow \mathscr {F}'|_{\operatorname{\mathcal{E}}'}$ denote natural transformations which exhibit $\mathscr {F}$ and $\mathscr {F}'$ as covariant transport representations of $U$ and $U'$, in the sense of Definition 7.4.1.8.