Remark 2.2.8.31 (The Universal Property of the Core). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is characterized by the following properties:
The $2$-category $\operatorname{\mathcal{C}}^{\simeq }$ is a $2$-groupoid (Remark 2.2.8.30).
For every $2$-groupoid $\operatorname{\mathcal{D}}$, every functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ factors (uniquely) through $\operatorname{\mathcal{C}}^{\simeq }$.