Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 2.1.4.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ and $\operatorname{\mathcal{D}}^{\operatorname{rev}}$ denote the same categories with the reversed nonunital monoidal structure (Example 2.1.3.5). Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ can be also regarded as a functor from $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ to $\operatorname{\mathcal{D}}^{\operatorname{rev}}$, which we will denote by $F^{\operatorname{rev}}$. There is a canonical bijection

\[ \xymatrix { \{ \text{Nonunital lax monoidal structures on $F$} \} \ar [d]^{\sim } \\ \{ \text{Nonunital lax monoidal structures on $F^{\operatorname{rev}}$} \} , } \]

which carries a nonunital lax monoidal structure $\mu $ to the nonunital lax monoidal structure $\mu ^{\operatorname{rev}}$ given by the formula $\mu ^{\operatorname{rev}}_{X,Y} = \mu _{Y,X}$. Using these bijections, we obtain a canonical isomorphism of categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$, which restricts to an isomorphism $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$.