Proposition 8.2.6.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the diagram (8.55) is an equivalence of couplings (in the sense of Exercise 8.2.2.7)
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Proposition 8.1.7.6 that the inclusion maps
\[ \rho _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \hookrightarrow \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}) \quad \quad \rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}) \]
are equivalences of $\infty $-categories. By virtue of Corollary 5.1.7.15, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism $\Xi $ induces a homotopy equivalence of Kan complexes
\[ \Xi _{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Cospan}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ). \]
We complete the proof by observing that $\Xi _{X,Y}$ fits into a commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [d] \\ \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \ar [r]^-{ \Xi _{X,Y} } & \operatorname{Cospan}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) } \]
where the left vertical map is the homotopy equivalence of Corollary 8.1.2.10, the right vertical map is the homotopy equivalence of Example 8.1.7.7, and the upper horizontal map is the homotopy equivalence of Proposition 4.6.5.10. $\square$