Kerodon

Notation 2.1.6.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, and let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be monoidal functors. We say that a natural transformation $\gamma : F \rightarrow F'$ is monoidal if it is monoidal when viewed as a natural transformation of lax monoidal functors (Definition 2.1.5.18). We let $\operatorname{Fun}^{\otimes }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the category whose objects are monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and whose morphisms are monoidal natural transformations. We regard $\operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ as a full subcategory of the category $\operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of Definition 2.1.5.18 (or as a non-full subcategory of the category $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of nonunital monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$).