Kerodon

Remark 9.3.1.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $n$ be an integer, and let $X \in \operatorname{\mathcal{C}}$ be an object whose image $F(X)$ is an $n$-truncated object of $\operatorname{\mathcal{D}}$. If the functor $F$ is essentially $(n+1)$-categorical (Definition 4.8.6.1), then $X$ is an $n$-truncated object of $\operatorname{\mathcal{C}}$ (see Proposition 3.5.9.13). In particular, if $F$ is fully faithful, then $X$ is an $n$-truncated object of $\operatorname{\mathcal{C}}$.