Notation 9.3.2.18 (The Heart of an $\infty $-Category). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We let $\operatorname{\mathcal{C}}^{\heartsuit }$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the discrete objects of $\operatorname{\mathcal{C}}$. We will refer to $\operatorname{\mathcal{C}}^{\heartsuit }$ as the heart of the $\infty $-category $\operatorname{\mathcal{C}}$.
Let $\mathrm{Disc}(\operatorname{\mathcal{C}})$ denote the homotopy category of $\operatorname{\mathcal{C}}^{\heartsuit }$. By construction, the $\infty $-category $\operatorname{\mathcal{C}}^{\heartsuit }$ is locally discrete, so Remark 9.3.2.10 guarantees that the comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ restricts to a trivial Kan fibration
For this reason, we will often abuse terminology by identifying the heart $\operatorname{\mathcal{C}}^{\heartsuit }$ with the ordinary category $\mathrm{Disc}(\operatorname{\mathcal{C}})$, which we also refer to as the heart of $\operatorname{\mathcal{C}}$.