Kerodon

Warning 8.4.1.13 (Failure of Transitivity). Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{E}}$ be full subcategories. Suppose that $\operatorname{\mathcal{C}}$ is a dense subcategory of $\operatorname{\mathcal{E}}$. Then $\operatorname{\mathcal{C}}$ is also a dense subcategory of $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{D}}$ is a dense subcategory of $\operatorname{\mathcal{E}}$ (see Corollary 7.3.8.8). Beware that the converse is false (Example 8.4.1.14).