Exercise 4.8.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $Q = \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ denote the collection of isomorphism classes of objects of $\operatorname{\mathcal{C}}$. For each object $X \in \operatorname{\mathcal{C}}$, let $[X] \in Q$ denote its isomorphism class. Show that:
There is a partial ordering $\leq _{Q}$ on the set $Q$, where $[X] \leq _{Q} [Y]$ if and only if there exists a morphism from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
There is a unique functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$ which carries each object $X \in \operatorname{\mathcal{C}}$ to the isomorphism class $[X] \in Q$.
The functor $F$ exhibits $\operatorname{N}_{\bullet }(Q)$ as a homotopy $0$-category of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.4.1.