Kerodon

Lemma 8.1.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, which we identify with a diagram $X: \operatorname{Tw}( \Lambda ^{n}_{0} ) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Assume that $n \geq 3$ and that the morphisms $X(0,0) \rightarrow X(0,1)$ and $X(1,n) \rightarrow X(0,n)$ are isomorphisms in $\operatorname{\mathcal{C}}$. Then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.