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

           and         

commute for every objects $A$ and $B$ of ${\mathcal {C}}$ (see Definition 11 in ).

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