Remark 9.3.2.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories, and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then:
If $F(X)$ is a discrete object of $\operatorname{\mathcal{D}}$, then $X$ is a discrete object of $\operatorname{\mathcal{C}}$.
If $F(X)$ is a subterminal object of $\operatorname{\mathcal{D}}$, then $X$ is a subterminal object of $\operatorname{\mathcal{C}}$.
In both cases, the converse holds if $F$ is an equivalence of $\infty $-categories.