Corollary 7.3.8.7 (Limits of Truncated Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then the collection of $n$-truncated objects of $\operatorname{\mathcal{C}}$ is closed under limits.
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Proof. Without loss of generality, we may assume that $n \geq -2$ (Remark 4.7.1.3). Let $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( (\operatorname{\partial \Delta }^{n+2})^{\triangleleft }, \operatorname{\mathcal{C}})$ be the diagonal map. By virtue of Remark 7.1.3.13, an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if $\delta (X)$ is a limit diagram. The desired result now follows from (the dual of) Corollary 7.3.8.6. $\square$