Remark 8.2.2.3 (044B): Functor Couplings

Remark 8.2.2.3 (Functor Couplings). 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. Proposition 8.2.2.2 asserts that the induced map

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

is also a coupling of $\infty $-categories. Moreover, it is characterized by a universal property: for every coupling of $\infty $-categories $\kappa : \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{B}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{B}}_{+}$, there is a canonical isomorphism of simplicial sets

\[ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{B}}, \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})) \simeq \operatorname{Fun}_{\pm }( \operatorname{\mathcal{B}}\times \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}), \]

where the right hand side is defined using the product coupling

\[ \operatorname{\mathcal{B}}\times \operatorname{\mathcal{C}}\xrightarrow { \kappa \times \lambda } (\operatorname{\mathcal{B}}_{-} \times \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} \times ( \operatorname{\mathcal{B}}_{+} \times \operatorname{\mathcal{C}}_{+} ). \]