Remark 2.2.3.6. Constructions 2.2.3.1 and 2.2.3.4 are analogous but not identical. At the level of $2$-morphisms, passage from a $2$-category $\operatorname{\mathcal{C}}$ to its opposite $\operatorname{\mathcal{C}}^{\operatorname{op}}$ reverses the order of horizontal composition, but preserves the order of vertical composition; passage from $\operatorname{\mathcal{C}}$ to its conjugate $\operatorname{\mathcal{C}}^{\operatorname{c}}$ preserves the order of horizontal composition and reverses the order of vertical composition. Following the notation of Warning 2.2.1.9, we have
\[ \delta ^{\operatorname{op}} \gamma ^{\operatorname{op}} = ( \delta \gamma )^{\operatorname{op}} \quad \quad \gamma ^{\operatorname{op}} \circ \gamma '^{\operatorname{op}} = (\gamma ' \circ \gamma )^{\operatorname{op}} \]
\[ \gamma ^{\operatorname{c}} \delta ^{\operatorname{c}} = ( \delta \gamma )^{\operatorname{c}} \quad \quad \gamma '^{\operatorname{c}} \circ \gamma ^{\operatorname{c}} = (\gamma ' \circ \gamma )^{\operatorname{c}}. \]