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Warning 4.5.5.12. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X'_{} \ar [d]^{q'} \ar [r]^-{f'} & X_{} \ar [d]^{q} \\ S'_{} \ar [r]^-{f} & S_{}, } \]

where $q$ is an isofibration. If $f$ is an equivalence of $\infty $-categories, then $f'$ is also an equivalence of $\infty $-categories (Corollary 4.5.2.23). Beware that if $f$ is merely assumed to be a categorical equivalence of simplicial sets, then it is not necessarily true that $f'$ is a categorical equivalence of simplicial sets.