Corollary 5.2.4.7. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d] & Y \ar [d] \\ X' \ar [r] & Y', } \]
where the vertical maps are categorical equivalences. Then the induced map $X \star _{Y} Y \rightarrow X' \star _{Y'} Y'$ is also a categorical equivalence of simplicial sets.