Warning 2.2.8.11. In the situation of Remark 2.2.8.10, it is not true that a lax functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ factors through the pith $\operatorname{Pith}(\operatorname{\mathcal{C}})$ (even when $\operatorname{\mathcal{D}}$ is a $(2,1)$-category): any lax functor which admits such a factorization is automatically a functor, by virtue of Remark 2.2.8.8.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$