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Proposition Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the functor $\rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories.

Proof of Proposition Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We wish to show that the comparison map $\rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$ of Construction Let $\operatorname{\mathcal{D}}$ be a simplicial set; we will show that composition with $\rho _{+}$ induces a bijection $\theta : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) )^{\simeq } )$.

Let $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ denote the projection map, and let $W$ be the collection of all edges $e$ of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ such that $\lambda _{+}(e)$ is a degenerate edge of $\operatorname{\mathcal{D}}$. Let $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}})[W^{-1}], \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ spanned by those diagrams $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ which carry each edge of $W$ to an isomorphism in $\operatorname{\mathcal{C}}$ (Notation Using Lemmas and, we can identify $\theta $ with the map $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}})[W^{-1}], \operatorname{\mathcal{C}})^{\simeq } )$ given by composition $\lambda _{+}$. To complete the proof, it will suffice to show that $\lambda _{+}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ with respect to $W$, in the sense of Definition This follows from Corollary, since the morphism $\lambda _{+}$ is universally localizing (Corollary $\square$