Remark 4.4.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category, and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\simeq }$ denote the core of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Then the core $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$ fits into a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{\simeq } \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\simeq }) \ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ). } \]