Variant 1.5.6.8. Let $S$ be a simplicial set. The following conditions are equivalent:
The simplicial set $S$ is isomorphic to the nerve of a category.
For every inner anodyne map of simplicial sets $i: A \hookrightarrow B$ and every map $f_0: A \rightarrow S$, there exists a unique map $f: B \rightarrow S$ such that $f_0 = f \circ i$.
Proof. Let us regard the simplicial set $S$ as fixed, and let $T$ be the collection of all morphisms of simplicial sets $i: A \rightarrow B$ for which the induced map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( B, S ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, S )$ is bijective. Then $T$ is weakly saturated (in the sense of Definition 1.5.4.12). It follows that $(2)$ is equivalent to the following a priori weaker assertion:
For every pair of integers $0 < i < n$, the map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_ i, S )$ is bijective.
The equivalence of $(1)$ and $(2')$ is the content of Proposition 1.3.4.1. $\square$