Example 5.2.5.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Combining Remark 5.2.5.4 with Theorem 5.2.2.20, we deduce that the following conditions are equivalent:
The morphism $U$ is a Kan fibration.
The morphism $U$ is a cocartesian fibration and the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 5.2.5.2 factors through the subcategory $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq } \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$.
The morphism $U$ is a cartesian fibration and the homotopy transport representation $\operatorname{hTr}'_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 5.2.5.7 factors through the subcategory $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq } \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$.
If these conditions are satisfied, then $\operatorname{hTr}'_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is given by the composition
where $\iota $ is the isomorphism which carries each morphism in $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq }$ to its inverse.