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Corollary 9.3.3.9 (Homotopy Invariance). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is $n$-truncated if and only if the image $F(f): F(X) \rightarrow F(Y)$ is $n$-truncated.

Proof. Using Corollary 4.6.4.19, we see that $F$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / F(Y) }$. The desired result now follows by combining Proposition 9.3.3.7 with Remark 9.3.1.7. $\square$