Remark 4.6.3.4. In the situation of Definition 4.6.3.3, condition $(2)$ is equivalent to the requirement that the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{i} \ar [d]^{i} & B \ar [d]^{s_0} \\ B \ar [r]^-{s_1} & \overline{B} } \]
commutes and is a pullback square (note that the morphisms $s_0$ and $s_1$ are automatically monomorphisms, since they are right inverse to the map $\pi : \overline{B} \rightarrow B$).