Definition 4.5.3.1. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. We say that $f$ is a categorical equivalence if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the induced functor $\operatorname{Fun}( Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ induces a bijection on isomorphism classes $\pi _0( \operatorname{Fun}(Y,\operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } )$.
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