# Kerodon

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Proposition 8.2.2.2. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ and $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+}$ be couplings of $\infty$-categories. Then the projection maps

$\Phi _{-}: \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \quad \quad \Phi _{+}: \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} )$

induce a left fibration

$( \Phi _{-}, \Phi _{+}): \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ).$

Proof. By construction, there is a pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r] \ar [d]^{ ( \Phi _{-}, \Phi _{+})} & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d]^{ \mu \circ } \\ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \ar [r]^-{ \circ \lambda } & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} ). }$

It will therefore suffice to show that the right vertical map is a left fibration (Remark 4.2.1.8), which follows from our assumption that $\mu$ is a left fibration (Corollary 4.2.5.2). $\square$