Example 11.10.2.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a fibration in sets (Definition 4.2.3.1). Then $U$ admits a unique cleavage, and the associated transport representation $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ is given by the composition
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow { \chi _{U} } \operatorname{Set}\hookrightarrow \mathbf{Cat}, \]
where $\chi _{U}$ is the transport representation of Construction 11.10.5.7 and the inclusion $\operatorname{Set}\hookrightarrow \mathbf{Cat}$ carries each set to the associated discrete category.