Kerodon

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

Remark 4.6.1.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of morphisms $f,g: X \rightarrow Y$ having the same source and target. Then $f$ and $g$ are homotopic (Definition 1.3.3.1) if and only if they belong to the same connected component of the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$: this follows from the characterization of Corollary 1.3.3.7. Consequently, we obtain a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \simeq \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$.