Remark 4.8.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. For every integer $n$, the following conditions are equivalent:
The morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated.
The left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ is $n$-truncated.
The right-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ is $n$-truncated.
This follows from Corollary 3.5.7.12, since the pinch inclusion maps
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookleftarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \]
are homotopy equivalences (Proposition 4.6.5.10).