Example 8.2.4.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, let $(\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ and $(\mu _{-},\mu _{+}): \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ be the twisted arrow couplings of Example 8.2.0.2, and let
\[ \operatorname{ev}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\quad \quad (F,C) \mapsto F(C) \]
be the evaluation functor. Passing to twisted arrow $\infty $-categories, we obtain a map
\[ \operatorname{Tw}(\operatorname{ev}): \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \times \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}}), \]
which we can identify with a functor $\widetilde{E}: \operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. By construction, the functor $\widetilde{E}$ fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \ar [d]^{\circ \lambda _{-}} & \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \ar [l] \ar [d]^{ \widetilde{T} } \ar [r] & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d]^{\circ \lambda _{+}} \\ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}^{\operatorname{op}} ) & \operatorname{Fun}( \operatorname{Tw}( \operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) ) \ar [l]_{\mu _{-} \circ } \ar [r]^-{\mu _{+} \circ } & \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}), } \]
and therefore determines a functor $E: \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. The commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \ar [r]^-{E} \ar [d] & \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) ) \ar [d]^{\Phi } \\ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r]^-{ \operatorname{id}\times \operatorname{id}} & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \]
exhibits the coupling $\Phi $ as represented by the identity functor $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.