Remark 4.2.1.8. The collections of left and right fibrations are closed under pullback. That is, suppose we are given a pullback diagram of simplicial sets

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

If $f$ is a left fibration, then $f'$ is also a left fibration. If $f$ is a right fibration, then $f'$ is a right fibration.