Remark 4.8.5.9. In the situation of Proposition 4.8.5.8, the comparison map $\operatorname{\mathcal{D}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} )$ is automatically an isofibration (Corollary 4.4.1.9). By virtue of Proposition 4.5.2.26, condition $(2)$ is equivalent to the following:
- $(2')$
The functor $F$ induces an equivalence of $\infty $-categories $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \times _{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} )} \operatorname{\mathcal{D}}$.
Note that the functor $F'$ is bijective on objects, and therefore essentially surjective. Using Theorem 4.6.2.21, we can reformulate $(2')$ as follows:
- $(2'')$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces a homotopy equivalence
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \times _{ \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) ) } \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ). \]