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Remark 8.2.2.4. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ and $\mu = (\mu _{-}, \mu _{+} ): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ be couplings of $\infty$-categories, and suppose we are given a morphism of couplings

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{F^{\operatorname{op}}_{-} \times F_{+}} & \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+}.}$

It follows from Proposition 8.2.1.7 that the functor $F$ carries $\lambda _{-}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to $\mu _{-}$-cocartesian morphisms of $\operatorname{\mathcal{D}}$, and $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to $\mu _{+}$-cocartesian morphisms of $\operatorname{\mathcal{D}}$.