monoidal natural transformation
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
A monoidal natural transformation is a natural transformation between monoidal functors that respects the monoidal structure.
Let and be two (braided monoidal categories). Let for be two (braided) lax monoidal functors .
Then a homomorphism between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is
- a natural transformation of the underlying functors
compatible with the product and the unit in that the following diagrams commute for all objects :