# 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. Applying Lemma 5.2.4.13, we see that the natural map

$\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^0 \star _{\Delta ^0} \Delta ^0 \simeq \Delta ^1$

is an inner fibration of simplicial sets. Since $\Delta ^1$ is an $\infty$-category, it follows that $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty$-category (Remark 4.1.1.9). $\square$