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Example 5.4.6.12 (Objects and Morphisms of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$). The inclusion of simplicial sets $\operatorname{\mathcal{QC}}\hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }$ induces a functor of $\infty $-categories $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$. The functor $\iota $ is bijective on vertices. In particular, we can identify the objects of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ with pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a (small) $\infty $-category and $C \in \operatorname{\mathcal{C}}$ is an object. However, it is not bijective on edges. Unwinding the definitions, we see that a morphism $\widetilde{F}$ from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ 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 a morphism in the $\infty $-category $\operatorname{\mathcal{D}}$. For every such pair $(F,\alpha )$, the following conditions are equivalent:

  • The morphism $\widetilde{F} = (F,\alpha )$ belongs to the image of the inclusion map $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.

  • The morphism $\alpha : F(C) \rightarrow D$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

  • The morphism $\widetilde{F}$ is $V$-cocartesian, where $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ is the cocartesian fibration of Proposition 5.4.6.11.