Monoidal natural transformation

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

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

are two lax monoidal functors between those categories.

A monoidal natural transformation

${\displaystyle \theta :(F,m)\to (G,n)}$

between those functors is a natural transformation ${\displaystyle \theta :F\to G}$between the underlying functors such that the diagrams

           and         

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

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

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