Proposition 5.5.2.6 (02S2)

Proposition 5.5.2.6. Let $U: \operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ be the left fibration of Proposition 5.5.2.2, and let

\[ \operatorname{hTr}_{ \operatorname{\mathcal{S}}_{\ast } / \operatorname{\mathcal{S}}}: \operatorname {h}\! \mathit{\operatorname{\mathcal{S}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{Kan}} \]

be the enriched homotopy transport representation of Variant 5.2.8.11. Then $\operatorname{hTr}_{ \operatorname{\mathcal{S}}_{\ast } / \operatorname{\mathcal{S}}}$ is homotopy inverse (as an $\operatorname {h}\! \mathit{\operatorname{Kan}}$-enriched functor) to the isomorphism $\operatorname {h}\! \mathit{\operatorname{Kan}} \simeq \operatorname {h}\! \mathit{\operatorname{\mathcal{S}}}$ of Remark 3.1.6.4. In particular, $\operatorname{hTr}_{\operatorname{\mathcal{S}}_{\ast } / \operatorname{\mathcal{S}}}$ is an equivalence of $\operatorname {h}\! \mathit{\operatorname{Kan}}$-enriched categories.

Proof. Apply Theorem 5.4.9.2 to the simplicial category $\operatorname{Kan}$. $\square$