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Remark 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.