Remark 4.2.1.7. The collection of left and right fibrations is closed under retracts. That is, suppose we are given a diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_{} \ar [r] \ar [d]^{f} & X'_{} \ar [d]^{f'} \ar [r] & X_{} \ar [d]^{f} \\ S_{} \ar [r] & S'_{} \ar [r] & S_{} } \]
where both horizontal compositions are the identity. If $f'$ is a left fibration, then $f$ is a left fibration. If $f'$ is a right fibration, then $f$ is a right fibration.