Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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

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