# Kerodon

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Proposition 5.5.3.6. Let $U: \operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ be the left fibration of Proposition 5.5.3.2, and let

$\operatorname{hTr}_{ \operatorname{\mathcal{S}}_{\ast } / \operatorname{\mathcal{S}}}: \mathrm{h} \mathit{\operatorname{\mathcal{S}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$

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 $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor) to the isomorphism $\mathrm{h} \mathit{\operatorname{Kan}} \simeq \mathrm{h} \mathit{\operatorname{\mathcal{S}}}$ of Remark 5.5.1.6. In particular, $\operatorname{hTr}_{\operatorname{\mathcal{S}}_{\ast } / \operatorname{\mathcal{S}}}$ is an equivalence of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories.

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