Kerodon

Warning 2.3.1.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{\mathcal{C}}^{\operatorname{c}}$ be the conjugate of $\operatorname{\mathcal{C}}$, obtained by reversing vertical composition (Construction 2.2.3.4). There is no simple relationship between Duskin nerves of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}^{\operatorname{c}}$ (since the operation $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{c}}$ is not functorial with respect to lax functors; see Warning 2.2.5.11).