Monoidal natural transformation

Template:Refimprove Suppose that $({𝒞},\otimes ,I)$ and $({𝒟},\bullet ,J)$ are two monoidal categories and

$(F,m):({𝒞},\otimes ,I)\to ({𝒟},\bullet ,J)$ and $(G,n):({𝒞},\otimes ,I)\to ({𝒟},\bullet ,J)$

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 ${𝒞}$.

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