Monoidal natural transformation
Encyclopedia
Suppose that 
and 
are two monoidal categories
and
and 
are two lax monoidal functors between those categories.
A monoidal natural transformation
between those functors is a natural transformation
between the underlying functors such that the diagrams
commute for every objects
and 
of 
.
A symmetric monoidal natural transformation is a monoidal natural transformation between symmetric monoidal functors.
and are two monoidal categoriesMonoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
and
and are two lax monoidal functors between those categories.
A monoidal natural transformation
between those functors is a natural transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
between the underlying functors such that the diagrams
and 
commute for every objects
and of .A symmetric monoidal natural transformation is a monoidal natural transformation between symmetric monoidal functors.