Remark 9.3.1.24. If $n \geq -1$, we can reformulate condition $(3)$ of Proposition 9.3.1.23 as follows:
- $(3')$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated. Moreover, the summand $\operatorname{Isom}_{\operatorname{\mathcal{C}}}(X,Y) \subseteq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ spanned by the isomorphisms from $X$ to $Y$ is $(n-1)$-truncated.
See Example 3.5.9.18.