Corollary 220.127.116.11. 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
where $u$ is a categorical equivalence by hypothesis and the vertical maps are categorical equivalences by Proposition 18.104.22.168. Using Remark 22.214.171.124, we conclude that $\Phi (u)$ is a categorical equivalence as well. $\square$