Remark 4.5.3.6. The collection of categorical equivalences is closed under retracts. That is, if there exists a commutative diagram of simplicial sets
\[ \xymatrix@C =40pt@R=40pt{ X \ar [r] \ar [d]^{f} & X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y \ar [r] & Y' \ar [r] & Y } \]
where the horizontal compositions are the identity and $f'$ is a categorical equivalence, then $f$ is also a categorical equivalence.