Kerodon

Remark 2.1.6.11 (Compatibility with Reversal). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor, and let $F^{\operatorname{rev}}: \operatorname{\mathcal{C}}^{\operatorname{rev}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{rev}}$ be as in Example 2.1.4.13. Then $F$ is a monoidal functor if and only if $F^{\operatorname{rev}}$ is a monoidal functor. This observation (and its counterpart for monoidal natural transformations) supplies an isomorphism of categories $\operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$.