# Kerodon

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Proposition 4.6.4.2. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category, and suppose we are given morphisms of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Then the projection map $\theta : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration of simplicial sets.

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

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d]^{\theta } & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \ar [d]^{\theta _0} \\ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}). }$

Since $\operatorname{\mathcal{E}}$ is an $\infty$-category, the restriction map $\theta _0$ is an isofibration of $\infty$-categories (Corollary 4.4.5.3). Invoking Remark 4.5.5.11, we conclude that $\theta$ is an isofibration of simplicial sets. $\square$