Monoidal natural transformation explained

Suppose that

( l C, ⊗ ,I) and ( l D,\bullet, J) are two monoidal categories and(F,m):( l C, ⊗ ,I)\to( l D,\bullet, J) and (G,n):( l C, ⊗ ,I)\to( l 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 l C .^[1]

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

References

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

Notes and References

1. Web site: Baez. John C.. Some Definitions Everyone Should Know. 2 December 2014.