Kerodon

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Definition 2.1.4.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories. Let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors equipped with nonunital lax monoidal structures $\mu$ and $\mu '$, respectively. We say that a natural transformation of functors $\gamma : F \rightarrow F'$ is nonunital monoidal if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagram

$\xymatrix@C =50pt@R=50pt{ F(X) \otimes F(Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ \gamma (X) \otimes \gamma (Y) } & F(X \otimes Y) \ar [d]^{\gamma (X \otimes Y)} \\ F'(X) \otimes F'(Y) \ar [r]^-{ \mu '_{X,Y}} & F'(X \otimes Y) }$

is commutative.

We let $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the category whose objects are nonunital lax monoidal functors $(F,\mu )$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, and whose morphisms are nonunital monoidal natural transformations, and we let $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the nonunital monoidal functors $(F,\mu )$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.