Kerodon

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Remark 4.5.2.7. Let $f: X \rightarrow Y$ be a categorical equivalence of simplicial sets. Then, for any simplicial set $K$, the induced map $f_ K: X \times K \rightarrow Y \times K$ is also a categorical equivalence of simplicial sets. To prove this, we must show that for every $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\theta : \operatorname{Fun}( Y \times K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X \times K, \operatorname{\mathcal{C}})$ induces a bijection on isomorphism classes of objects. This follows from our assumption that $f$ is a categorical equivalence, since $\theta $ can be identified with the map $\operatorname{Fun}(Y, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}(X, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) )$ given by precomposition with $f$.