Kerodon

Example 2.2.3.7. Let $\operatorname{\mathcal{C}}$ be an ordinary category, which we regard as a $2$-category having only identity $2$-morphisms (Example 2.2.0.6). Then the opposite $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ of Construction 2.2.3.1 coincides with the opposite of $\operatorname{\mathcal{C}}$ as an ordinary category (which we can again regard as a $2$-category having only identity morphisms). The conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ of Construction 2.2.3.4 can be identified with $\operatorname{\mathcal{C}}$ itself.