# Kerodon

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Example 5.4.6.12 (Objects and Morphisms of $\operatorname{\mathcal{QC}}^{\operatorname{lax}}_{\ast }$). The inclusion of simplicial sets $\operatorname{\mathcal{QC}}\hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }$ induces a functor of $\infty$-categories $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$. The functor $\iota$ is bijective on vertices. In particular, we can identify the objects of $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ 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 $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ 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 $\iota : \operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$.

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

• The morphism $\widetilde{F}$ is $U_0$-cocartesian, where $U_0: \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \rightarrow \operatorname{\mathcal{QC}}$ is the cocartesian fibration of Proposition 5.4.6.11.