# Kerodon

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Corollary 4.6.4.12. Let $\operatorname{\mathcal{D}}$ be an $\infty$-category containing an object $X$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration, and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. Unwinding the definition, we have pullback diagrams

$\xymatrix@R =50pt@C=50pt{ \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\ar [r] \ar [d] & \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}& \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}. }$

The desired result now follows by combining Proposition 4.6.4.11 with Remark 4.2.1.8. $\square$