Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 2.2.5.10 (Functors on Conjugate $2$-Categories). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories, and let $\operatorname{\mathcal{C}}^{\operatorname{c}}$ and $\operatorname{\mathcal{D}}^{\operatorname{c}}$ denote their conjugates (Construction 2.2.3.4). Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ induces a functor $F^{\operatorname{c}}: \operatorname{\mathcal{C}}^{\operatorname{c}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{c}}$, given explicitly by the formulae

$F^{\operatorname{c}} (X^{\operatorname{c}} ) = F(X)^{\operatorname{c}} \quad \quad F^{\operatorname{c}}(f^{\operatorname{c}} ) = F(f)^{\operatorname{c}} \quad \quad F^{\operatorname{c}}( \gamma ^{\operatorname{c}} ) = F(\gamma )^{\operatorname{c}}$
$\epsilon _{X^{\operatorname{c}}} = (\epsilon ^{-1}_{X})^{\operatorname{c}} \quad \quad \mu _{ g^{\operatorname{c}}, f^{\operatorname{c}} } = (\mu ^{-1}_{g,f})^{\operatorname{c}}.$

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

$\operatorname{2Cat}_{\operatorname{Str}} \simeq \operatorname{2Cat}_{\operatorname{Str}} \quad \quad \operatorname{2Cat}\simeq \operatorname{2Cat}$