Proposition 8.1.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the simplicial set $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ is an $\infty $-category.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\sigma $ be a $2$-simplex of $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$, which we identify with a diagram
\[ \xymatrix@C =50pt@R=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dl]_{\sim } \ar [dr] & & X_{2,2} \ar [dl]_{\sim } \\ & X_{0,1} \ar [dr] & & X_{1,2} \ar [dl]_{\sim } & \\ & & X_{0,2} & & } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$ where the leftward-directed morphisms are isomorphisms. Using Corollary 7.6.2.27, we deduce that the inner region is a pushout square in $\operatorname{\mathcal{C}}$. It follows that $\sigma $ is automatically thin when regarded as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ (Proposition 8.1.4.2), and therefore also when regarded as a $2$-simplex of $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ (Remark 8.1.6.4). To complete the proof, it will suffice to show that every diagram $\Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ can be extended to a $2$-simplex of $\operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$ (see Example 2.3.2.4). Using Lemma 8.1.4.4, we can restate this as follows: for every pair of morphisms $f: X_{1,1} \rightarrow X_{1,2}$ and $u: X_{1,1} \rightarrow X_{0,1}$ of $\operatorname{\mathcal{C}}$ where $u$ is an isomorphism, there exists a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ & X_{1,1} \ar [dl]_{u} \ar [dr]^{f} & \\ X_{0,1} \ar [dr] & & X_{1,2} \ar [dl]_{v} \\ & X_{0,2} & } \]
where $v$ is also an isomorphism. This follows immediately from the definitions (or from Corollary 4.4.5.9). $\square$