Remark 4.5.4.13. Suppose we are given a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{g} & A_0 \ar [d]^{g'} \\ A_1 \ar [r] & A_{01}, } \]
where $f$ is a monomorphism. If $g$ is a categorical equivalence, then $g'$ is also a categorical equivalence. This follows by combining Example 4.5.4.12 with Proposition 4.5.4.10.