Definition 2.1.5.18. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be lax monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. We will say that a natural transformation $\gamma : F \rightarrow F'$ is monoidal if it satisfies the following pair of conditions:
The natural transformation $\gamma $ is nonunital monoidal, in the sense of Definition 2.1.4.10. That is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagram
\[ \xymatrix@C =50pt@R=50pt{ F(X) \otimes F(Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ \gamma (X) \otimes \gamma (Y) } & F(X \otimes Y) \ar [d]^{\gamma (X \otimes Y)} \\ F'(X) \otimes F'(Y) \ar [r]^-{ \mu '_{X,Y}} & F'(X \otimes Y) } \]commutes, where $\mu $ and $\mu '$ are the tensor constraints of $F$ and $F'$, respectively.
The unit of $F'$ is equal to the composition $\mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow {\epsilon } F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { \gamma ( \mathbf{1}_{\operatorname{\mathcal{C}}} )} F'(\mathbf{1}_{\operatorname{\mathcal{C}}} )$, where $\epsilon $ is the unit of $F$.
We let $\operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the category whose objects are lax monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and whose morphisms are monoidal natural transformations, which we regard as a (non-full) subcategory of the category $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ introduced in Definition 2.1.4.10.