Proposition 8.6.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the morphism $\Xi : \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} )$ is an equivalence of $\infty $-categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $V: \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be the projection map. It follows from Lemma 8.6.3.8 (and Corollary 8.1.1.14) that $V$ is a cartesian fibration, and that a morphism in $\operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} )$ is $V$-cartesian if and only if its image under the composite map
\[ \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \hookrightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \xrightarrow { \operatorname{Cospan}( \lambda _{+} ) } \operatorname{Cospan}(\operatorname{\mathcal{C}}) \]
is contained in $\operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}})$. Recall that the evaluation functor $\operatorname{ev}_0: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is also a cartesian fibration, and that a morphism $f$ of $\operatorname{Fun}( \Delta ^1,\operatorname{\mathcal{C}})$ is $\operatorname{ev}_0$-cartesian if and only if $\operatorname{ev}_1(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$ (Corollary 5.3.7.3). Invoking Remark 8.6.3.16, we obtain a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \ar [rr]^{ \Xi } \ar [dr]^{ \operatorname{ev}_0 } & & \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \ar [dl]_{V} \\ & \operatorname{\mathcal{C}}, & } \]
where $\Xi $ carries $\operatorname{ev}_0$-cartesian morphisms of the $\infty $-category $\operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})$ to $V$-cartesian morphisms of the $\infty $-category $\operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{C}}^{\operatorname{op}} )$. By virtue of Proposition 5.1.7.15, it will suffice to show that for each object $X \in \operatorname{\mathcal{C}}$, the functor $\Xi $ restricts to an equivalence of fibers
\[ \Xi _{X}: \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \rightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}} \operatorname{Cospan}^{\dagger }( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}} ) \simeq \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) ). \]
We conclude by observing that $\Xi _{X}$ fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [d] \ar [r] & \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d]^{\rho _{+}} \\ \{ X\} \otimes _{\operatorname{\mathcal{C}}} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \ar [r]^-{ \Xi _{X} } & \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) ) } \]
where the left vertical map is the equivalence of Corollary 4.6.4.18, the right vertical map is the equivalence of Proposition 8.1.7.6, and the upper horizontal map is the equivalence of Proposition 8.1.2.9. $\square$