Remark 4.5.5.10. The collection of isofibrations is closed under retracts. That is, given a diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_{} \ar [r] \ar [d]^{q} & X'_{} \ar [d]^{q'} \ar [r] & X_{} \ar [d]^{q} \\ S_{} \ar [r] & S'_{} \ar [r] & S_{} } \]
where both horizontal compositions are the identity, if $q'$ is an isofibration, then so is $q$.