Proposition 4.7.4.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}_0$ denote the homotopy category of the heart $\operatorname{\mathcal{C}}^{\heartsuit }$. Then the canonical map $U: \operatorname{\mathcal{C}}^{\heartsuit } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ is a trivial Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix an integer $m \geq 0$; we wish to show that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}^{\heartsuit } \ar [d] \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 ). } \]
For $m \leq 1$, this is immediate (since the right vertical map is bijective on vertices and surjective on edges). For $m = 2$, it follows from the definition of the composition law on the homotopy category $\operatorname{\mathcal{C}}_0$ (see Notation 1.4.4.3). For $m \geq 3$, it follows from the characterization of discrete objects given in Remark 4.7.4.20. $\square$