Remark 2.2.5.9 (0094): Functors on Opposite $2$-Categories

Remark 2.2.5.9 (Functors on Opposite $2$-Categories). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories, and let $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\operatorname{\mathcal{D}}^{\operatorname{op}}$ denote their opposites (Construction 2.2.3.1). Then every lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ induces a lax functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$, given explicitly by the formulae

\[ F^{\operatorname{op}} (X^{\operatorname{op}} ) = F(X)^{\operatorname{op}} \quad \quad F^{\operatorname{op}}(f^{\operatorname{op}} ) = F(f)^{\operatorname{op}} \quad \quad F^{\operatorname{op}}( \gamma ^{\operatorname{op}} ) = F(\gamma )^{\operatorname{op}} \]

\[ \epsilon _{X^{\operatorname{op}}} = (\epsilon _{X})^{\operatorname{op}} \quad \quad \mu _{ g^{\operatorname{op}}, f^{\operatorname{op}} } = (\mu _{f,g})^{\operatorname{op}}. \]

In this case, $F$ is a functor if and only if $F^{\operatorname{op}}$ is a functor, and a strict functor if and only if $F^{\operatorname{op}}$ is a strict functor. This operation is compatible with composition, and therefore induces equivalences of categories

\[ \operatorname{2Cat}_{\operatorname{Str}} \simeq \operatorname{2Cat}_{\operatorname{Str}} \quad \quad \operatorname{2Cat}\simeq \operatorname{2Cat}\quad \quad \operatorname{2Cat}_{\operatorname{Lax}} \simeq \operatorname{2Cat}_{\operatorname{Lax}} \quad \quad \operatorname{2Cat}_{\operatorname{ULax}} \simeq \operatorname{2Cat}_{\operatorname{ULax}}. \]