# Kerodon

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Lemma 9.5.0.37. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. If $\operatorname{\mathcal{E}}$ is an $\infty$-category, then the natural 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.

Proof. By construction, we have a pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d] & \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{E}}\ar [r]^-{\rho } & \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}, }$

where $\rho$ is an inner fibration by virtue of Lemma 5.2.4.12. $\square$