Kerodon

Proof of Proposition 8.1.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. By virtue of Lemma 8.1.4.7 and Corollary 8.1.4.3, the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ satisfies conditions $(2)$ and $(3)$ of Definition 5.4.1.1. Since $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is isomorphic to $\operatorname{Cospan}(\operatorname{\mathcal{C}})^{\operatorname{op}}$ (Remark 8.1.3.4), it also satisfies condition $(4)$ of Definition 5.4.1.1. It follows that $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category if and only if it satisfies the following condition:

$(\ast )$

Every morphism of simplicial sets $\Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ can be extended to a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.

Using Lemma 8.1.4.4, we can rewrite condition $(\ast )$ as follows:

$(\ast ')$

For every diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [dr] & & Y \ar [dl] \ar [dr] & & Z \ar [dl] \\ & B & & C & } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$, there exists a pushout of $B$ and $C$ along $Y$.

It is clear that if the $\infty $-category $\operatorname{\mathcal{C}}$ admits pushouts, then it satisfies condition $(\ast ')$. The converse follows by applying condition $(\ast ')$ to diagrams of the form