Kerodon

Warning 5.5.6.7. The coslice comparison functor $U: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) \rightarrow \operatorname{\mathcal{QC}}_{\ast }$ of Proposition 5.5.6.6 is bijective on vertices: objects of either $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } )$ and $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is an $\infty $-category and $C$ is an object of $\operatorname{\mathcal{C}}$. However, it is not bijective on edges (and is therefore not an isomorphism of simplicial sets). If $(\operatorname{\mathcal{C}},C)$ and $(\operatorname{\mathcal{D}},D)$ are objects of $\operatorname{\mathcal{QC}}_{\ast }$, then a morphism from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with a pair $(F, \alpha )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories and $\alpha : F(C) \rightarrow D$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. The pair $(F, \alpha )$ belongs to the image of $U$ if and only if the isomorphism $\alpha $ is a degenerate edge of $\operatorname{\mathcal{D}}$ (which guarantees in particular that $F(C) = D$).