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Example 5.5.6.3 (Objects and Morphisms of $\operatorname{\mathcal{QC}}_{\ast }$). The low-dimensional simplices of the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ are easy to describe:

• The objects of $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a (small) $\infty $-category and $C \in \operatorname{\mathcal{C}}$ is an object (which we identify with the morphism $\Delta ^{0} \rightarrow \operatorname{\mathcal{C}}$ taking the value $C$).

• Let $(\operatorname{\mathcal{C}},C)$ and $(\operatorname{\mathcal{D}},D)$ be objects of $\operatorname{\mathcal{QC}}_{\ast }$. 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 $\operatorname{\mathcal{D}}$.

Example 5.5.6.12 (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 5.5.6.11.

Example 5.5.6.17 ($2$-Simplices of $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$). By virtue of Example 5.5.6.12, a morphism of simplicial sets $\sigma _0: \operatorname{\partial \Delta }^2 \rightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ can be identified with the following data:

• A collection of $\infty $-categories $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ equipped with distinguished objects $C \in \operatorname{\mathcal{C}}$, $D \in \operatorname{\mathcal{D}}$, and $E \in \operatorname{\mathcal{E}}$.

• A collection of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, and $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$.

• A collection of morphisms $\alpha : F(C) \rightarrow D$, $\beta : G(D) \rightarrow E$, and $\gamma : H(C) \rightarrow E$ in the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$.

Unwinding the definitions, we see that extending $\sigma _0$ to a $2$-simplex $\sigma $ of $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ is equivalent to choosing a natural transformation of functors $\mu : (G \circ F) \rightarrow H$ and a morphism of simplicial sets $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{\mathcal{E}}$ whose restriction to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is indicated in the diagram

Moreover:

• The $2$-simplex $\sigma $ belongs to the image of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}} \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ if and only if $\mu : G \circ F \rightarrow H$ is an isomorphism in the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.

• The $2$-simplex $\sigma $ belongs to the image of $\operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ if and only if $\mu $, $\alpha $, $\beta $, and $\gamma $ are all isomorphisms.

• The $2$-simplex $\sigma $ belongs to the image of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ if and only if $\mu $, $\alpha $, $\beta $, and $\gamma $ are identity morphisms (so that $H = G \circ F$, $D = F(C)$, and $E = G(D)$) and the morphism $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{\mathcal{E}}$ is constant.