Kerodon

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Corollary 9.2.3.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose that $\operatorname{\mathcal{C}}$ admits pullbacks and that the functor $F$ preserves pullbacks. Then, for every integer $n$, the functor $F$ carries $n$-truncated morphisms of $\operatorname{\mathcal{C}}$ to $n$-truncated morphisms of $\operatorname{\mathcal{D}}$.

Proof. For $n \leq -2$, a morphism is $n$-truncated if and only if it is an isomorphism (Example 9.2.3.5), so the desired result follows from Remark 1.5.1.6. The general case follows by induction on $n$, using Corollary 9.2.3.19. $\square$