Example 5.6.2.17. Let $\operatorname{\mathcal{E}}$ be a simplicial set, which we identify with the morphism of simplicial sets $\Delta ^{0} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$ taking the value $\operatorname{\mathcal{E}}$. Then the simplicial set $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ can be identified with the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$. In particular, Construction 4.6.8.3 supplies a comparison morphism
If $\operatorname{\mathcal{E}}$ is an $\infty $-category, then $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$ is also an $\infty $-category, and the comparison morphism $\rho $ is an equivalence of $\infty $-categories (Theorem 4.6.8.9). Beware that $\theta _{\operatorname{\mathcal{E}}}$ is generally not an isomorphism (though it is always a monomorphism which is bijective on simplices of dimension $\leq 1$). For example, Example 5.6.2.16 implies that $2$-simplices of $\int _{\Delta ^0} \operatorname{\mathcal{E}}$ can be identified with morphisms of simplicial sets $\rho : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$ for which the restriction $\rho |_{ \Delta ^1 \times \{ 0\} }$ is a degenerate edge of $\operatorname{\mathcal{E}}$, as indicated in the diagram
The corresponding $2$-simplex of $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ belongs to the image of $\theta _{\operatorname{\mathcal{E}}}$ if and only if $\sigma $ is a left-degenerate $2$-simplex of $\operatorname{\mathcal{E}}$ (in which case it is given by $\theta _{\operatorname{\mathcal{E}}}(\tau )$).