# Kerodon

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Notation 8.1.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Proposition 4.6.5.9 and Corollary 8.1.2.6 supply homotopy equivalences of Kan complexes

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookleftarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} .$

Passing to homotopy, we obtain an isomorphism

$\alpha _{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\sim } \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\}$

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.