Construction 8.1.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We let $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso} }(\operatorname{\mathcal{C}})$ denote the simplicial subset of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ whose $n$-simplices are diagrams
\[ \xymatrix@R =40pt@C=20pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl]_{\sim } & \cdots & X_{n-1,n-1} \ar [dr] \ar [dl]_{\sim } & & X_{n,n} \ar [dl]_{\sim } \\ & \cdots \ar [dr] & & \cdots \ar [dr] \ar [dl]_{\sim } & & \cdots \ar [dl]_{\sim } & \\ & & X_{0,n-1} \ar [dr] & & X_{1,n} \ar [dl]_{\sim } & & \\ & & & X_{0,n}, & & & \\ } \]
where each of the leftward-directed arrows is an isomorphism in $\operatorname{\mathcal{C}}$.