Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 5.2.4.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of $\infty $-categories. Then the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is an $\infty $-category.

Proof of Proposition 5.2.4.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. By virtue of Lemma 5.2.4.13, the tautological map $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an inner fibration of simplicial sets. Since $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an $\infty $-category (Proposition 4.3.3.22), it follows that $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty $-category (Remark 4.1.1.9). $\square$