Kerodon

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Definition 4.5.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. We say that a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is homotopy inverse to $F$ if the isomorphism class $[G]$ is an inverse to $[F]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$: that is, if $G \circ F$ and $F \circ G$ are isomorphic to the identity functors $\operatorname{id}_{\operatorname{\mathcal{C}}}$ and $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as objects of the $\infty$-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ and $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, respectively. We will say that $F$ is an equivalence of $\infty$-categories if $[F]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$: that is, if $F$ admits a homotopy inverse $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. We say that $\infty$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are equivalent if there exists an equivalence from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.