# Kerodon

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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). $\square$