Kerodon

Remark 2.1.6.12 (Opposite Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital monoidal functor, and let $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ be the induced nonunital monoidal functor on opposite categories (Example 2.1.4.14). Then $F$ is a monoidal functor if and only if $F^{\operatorname{op}}$ is a monoidal functor. This observation (and its counterpart for monoidal natural transformations) supplies an isomorphism of categories $\operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )$.