Remark 2.2.8.10 (The Universal Property of the Pith). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is characterized (up to isomorphism) by the following properties:
The pith $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category.
For every $(2,1)$-category $\operatorname{\mathcal{D}}$, every functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ factors (uniquely) through $\operatorname{Pith}(\operatorname{\mathcal{C}})$.