Remark 8.1.6.9. In the situation of Proposition 8.1.6.7, let $\sigma $ be an $n$-simplex of the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, corresponding to a diagram
8.18
\begin{equation} \begin{gathered}\label{pith-of-cospan} \xymatrix@R =40pt@C=20pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & \cdots & X_{n-1,n-1} \ar [dr] \ar [dl] & & X_{n,n} \ar [dl] \\ & \cdots \ar [dr] & & \cdots \ar [dr] \ar [dl] & & \cdots \ar [dl] & \\ & & X_{0,n-1} \ar [dr] & & X_{1,n} \ar [dl] & & \\ & & & X_{0,n} & & & \\ } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $\sigma $ is contained in the pith $\operatorname{Pith}( \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}}) )$ if and only if each of the rectangular regions in the diagram (8.18) is a pushout square in the $\infty $-category $\operatorname{\mathcal{C}}$. This follows from the thinness criterion of Corollary 8.1.6.8 (together with Proposition 7.6.2.28).