Remark 8.1.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a morphism of simplicial sets $\varphi : \operatorname{Tw}( \Delta ^2 ) \rightarrow \operatorname{\mathcal{C}}$, which we display as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dl] \ar [dr] & & X_{2,2} \ar [dl] \\ & X_{0,1} \ar [dr] & & X_{1,2} \ar [dl] & \\ & & X_{0,2}. & & } \]
The proof of Lemma 8.1.4.4 shows that $\varphi $ is a colimit diagram if and only if the inner region is a pushout square.