Example 9.3.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$. Then:
If the object $X$ is subterminal and $f$ is a monomorphism, then the object $X_0$ is also subterminal.
If the object $X_0$ is subterminal and the object $X$ is discrete, then $f$ is a monomorphism.
In particular, if $X$ is subterminal, then $f$ is a monomorphism if and only if $X_0$ is subterminal. See Proposition 9.3.3.14.