Kerodon

Example 4.6.7.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denote their join (Construction 4.3.3.13). Then $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is also an $\infty $-category (Corollary 4.3.3.25). It follows from Example 4.6.1.6 that if $X$ is an initial object of $\operatorname{\mathcal{C}}$, then it is also initial when regarded as an object of $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$. Similarly, if $Y$ is a final object of $\operatorname{\mathcal{D}}$, then it is also final when regarded as an object of $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$.