# Kerodon

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

Example 2.1.4.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and regard the opposite categories $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\operatorname{\mathcal{D}}^{\operatorname{op}}$ as equipped with the nonunital monoidal structures of Example 2.1.3.4. Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ determines a functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$. There is a canonical bijection

$\{ \text{Nonunital monoidal structures on F} \} \simeq \{ \text{Nonunital monoidal structures on F^{\operatorname{op}}} \} ,$

which carries a nonunital monoidal structure $\mu$ on $F$ to a nonunital monoidal structure $\mu '$ on $F^{\operatorname{op}}$, given concretely by $\mu '_{X,Y} = \mu _{X,Y}^{-1}$. Using these bijections, we obtain a canonical isomorphism of categories $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )$.