# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Warning 8.1.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Our proof of Corollary 8.1.2.13 shows that the isomorphism $\alpha _{X,Y}: \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \rightarrow H(X,Y)$ is compatible with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment in the second variable. Beware that things are a bit more subtle if we wish to view $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ and $H(X,Y)$ as $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors of the first variable. The functor $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}$ is defined using the enrichment of the category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, and can therefore be viewed an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

$(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}.$

On the other hand, the functor $H$ is defined as the enriched homotopy transport representation of the left fibration $(\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$, which is an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

$\mathrm{h} \mathit{( \operatorname{\mathcal{C}}^{\operatorname{op}})} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}.$

The $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories $( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )^{\operatorname{op}}$ and $\mathrm{h} \mathit{ ( \operatorname{\mathcal{C}}^{\operatorname{op}} )}$ are a priori different objects: to a pair of objects $X,Y \in \operatorname{\mathcal{C}}$, they assign morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}}$, respectively. It is possible to address this point (since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}}$ are canonically isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$), but we will not pursue the matter here.