Kerodon

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

Remark 5.5.4.5. Let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1). Then there is a tautological comparison map $\operatorname{\mathcal{QC}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{QCat}} )$, which carries each $\infty $-category $\operatorname{\mathcal{C}}$ to itself and each functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to its isomorphism class $[F] \in \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )$. This functor induces an isomorphism of homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \simeq \mathrm{h} \mathit{\operatorname{QCat}}$ (see Proposition 2.4.6.8).