Kerodon

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

Remark 5.4.6.13. The inclusion map $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ is an isomorphism from $\operatorname{\mathcal{QC}}_{\ast }$ to the (non-full) subcategory of $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ spanned by those morphisms which satisfy the conditions of Example 5.4.6.12. In other words, the projection map $\operatorname{\mathcal{QC}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}$ is the underlying left fibration of the cocartesian fibration $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \rightarrow \operatorname{\mathcal{QC}}$ (see Corollary 5.3.7.11).