Proposition 4.5.5.4. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then $i$ is a categorical equivalence if and only if the following condition is satisfied:
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories. Then every lifting problem
has a solution.
Proof. Assume that condition $(\ast )$ is satisfied; we will show that the morphism $i: A \hookrightarrow B$ is a categorical equivalence of simplicial sets (the converse follows from Proposition 4.5.5.1). Fix an $\infty $-category $\operatorname{\mathcal{E}}$; we wish to show that precomposition with $i$ induces a bijection $\theta : \pi _0( \operatorname{Fun}(B, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(A, \operatorname{\mathcal{E}})^{\simeq } )$. The surjectivity of $\theta $ follows by applying condition $(\ast )$ to the isofibration $\operatorname{\mathcal{E}}\rightarrow \Delta ^0$, and the injectivity of $\theta $ follows by applying $\theta $ to the isofibration $\operatorname{Isom}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}\times \operatorname{\mathcal{E}}$ of Corollary 4.4.5.5. $\square$