Remark 4.2.1.9. Let $f: X \rightarrow S$ be a map of simplicial sets. Suppose that, for every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is a left fibration (right fibration). Then $f$ is a left fibration (right fibration). Consequently, if we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X'_{} \ar [d]^{f'} \ar [r] & X_{} \ar [d]^{f} \\ S'_{} \ar [r]^-{g} & S_{} } \]

where $g$ is surjective and $f'$ is a left fibration (right fibration), then $f$ is also a left fibration (right fibration).