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Remark Constructions and 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, we have

\[ \delta ^{\operatorname{op}} \gamma ^{\operatorname{op}} = ( \delta \gamma )^{\operatorname{op}} \quad \quad \gamma ^{\operatorname{op}} \circ \gamma '^{\operatorname{op}} = (\gamma ' \circ \gamma )^{\operatorname{op}} \]
\[ \gamma ^{\operatorname{c}} \delta ^{\operatorname{c}} = ( \delta \gamma )^{\operatorname{c}} \quad \quad \gamma '^{\operatorname{c}} \circ \gamma ^{\operatorname{c}} = (\gamma ' \circ \gamma )^{\operatorname{c}}. \]