Proposition 1.5.6.7. Let $S$ be a simplicial set. The following conditions are equivalent:
The simplicial set $S$ is an $\infty $-category.
For every inner anodyne map of simplicial sets $i: A \hookrightarrow B$ and every map $f_0: A \rightarrow S$, there exists a map $f: B \rightarrow S$ such that $f_0 = f \circ i$.
Proof. The implication $(2) \Rightarrow (1)$ is immediate (since every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is inner anodyne). Conversely, if $(1)$ is satisfied, then every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is weakly left orthogonal to the projection map $p: S \rightarrow \Delta ^0$. It then follows from Remark 1.5.4.14 that every inner anodyne map is weakly left orthogonal to $p$. $\square$