Warning 11.10.5.9. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a fibration in sets and let $\chi _{U}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ be the transport representation associated to $U$ (Construction 11.10.5.7). If $U$ is also an opfibration in sets, then Variant 11.10.5.2 defines a functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ which we also refer to as the transport representation associated to $U$; to avoid confusion, let us temporarily denote the latter functor by $\operatorname{Tr}'_{U}$. The functors $\chi _{U}$ and $\operatorname{Tr}'_{U}$ are essentially interchangeable data: by virtue of Remark 11.10.5.4, the functor $\chi _ U$ factors through the (non-full) subcategory $\operatorname{Set}^{\simeq } \subseteq \operatorname{Set}$ spanned by the bijections of finite sets, and the functor $\operatorname{Tr}'_{U}$ is given by the composition
where $\iota : (\operatorname{Set}^{\simeq })^{\operatorname{op}} \rightarrow \operatorname{Set}^{\simeq }$ is the isomorphism of categories which carries every bijection $u: X \rightarrow Y$ to the inverse bijection $u^{-1}: Y \rightarrow X$.