Remark 1.3.4.4. The characterization of Proposition 1.3.4.1 has many variants. For example, one can replace condition $(\ast ')$ by the following a priori weaker condition:
- $(\ast '_{0})$
For every $n \geq 2$ and every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{1} \rightarrow S$, there is a unique $n$-simplex $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$ satisfying $\sigma _0 = \sigma |_{ \Lambda ^{n}_{1} }$.