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Corollary 4.6.7.25. Let $u: K \rightarrow K'$ be a categorical equivalence of simplicial sets. Then the induced map $\Phi (u): \Phi (K) \rightarrow \Phi (K')$ is also a categorial equivalence of simplicial sets.

Proof. We have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \Phi (K) \ar [r]^-{ \Phi (u) } \ar [d]^{\rho _ K} & \Phi (K') \ar [d]^{ \rho _{K'} } \\ K \ar [r]^-{u} & K' } \]

where $u$ is a categorical equivalence by hypothesis and the vertical maps are categorical equivalences by Proposition 4.6.7.24. Using Remark 4.5.3.5, we conclude that $\Phi (u)$ is a categorical equivalence as well. $\square$