Kerodon

Remark 5.5.6.16. The inclusion map $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ is an isomorphism from $\operatorname{\mathcal{QC}}_{\ast }$ to the (non-full) subcategory of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ spanned by those morphisms which satisfy the conditions of Example 5.5.6.12. In other words, the projection map $\operatorname{\mathcal{QC}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}$ is the underlying left fibration of the cocartesian fibration $\operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ (see Corollary 5.4.7.12).