# Kerodon

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Corollary 8.1.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f,f': X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$. Then $f$ and $f'$ are homotopic (in the sense of Definition 1.3.3.1) if and only they belong to the same connected component of the Kan complex $\{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\}$. Consequently, we have a canonical isomorphism of sets

$\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \simeq \pi _0( \{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} ).$