Remark 4.8.3.13. In the case $m = 0$, the formulation of Proposition 4.8.3.12 requires a slight modification. The restriction map $\theta _0$ is injective if and only if $\operatorname{\mathcal{C}}$ satisfies the following pair of conditions:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is minimal in dimension $0$: that is, if $X$ and $Y$ are isomorphic objects of $\operatorname{\mathcal{C}}$, then $X =Y$.
- $(2')$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an isomorphism from $X$ to $Y$.
Note that condition $(2')$ is stronger than condition $(2)$ of Proposition 4.8.3.12, which demands only that there exists a morphism from $X$ to $Y$.