Monoidal natural transformation

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.

References

• Perrone, Paolo (2024). Starting Category Theory. World Scientific. doi:10.1142/9789811286018_0005. ISBN 978-981-12-8600-1.