Monoidal natural transformation

Suppose that $$(\mathcal C,\otimes,I)$$ and $$(\mathcal D,\bullet, J)$$ are two monoidal categories and
 * $$(F,m):(\mathcal C,\otimes,I)\to(\mathcal D,\bullet, J)$$ and $$(G,n):(\mathcal C,\otimes,I)\to(\mathcal D,\bullet, J)$$

are two lax monoidal functors between those categories.

A monoidal natural transformation
 * $$\theta:(F,m) \to (G,n)$$

between those functors is a natural transformation $$\theta:F \to G$$ between the underlying functors such that the diagrams
 * [[File:Monoidal natural transformation multiplication.svg]]     and      [[File:Monoidal natural transformation unit.svg]]

commute for every objects $$A$$ and $$B$$ of $$\mathcal C$$.

A symmetric monoidal natural transformation is a monoidal natural transformation between symmetric monoidal functors.