Suppose that
and
are two
monoidal categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
and
:
and
are two
lax monoidal functors between those categories.
A monoidal natural transformation
:
between those functors is a
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
between the underlying functors such that the diagrams
: and
commute for every objects
and
of
(see Definition 11 in
).
A symmetric monoidal natural transformation is a monoidal natural transformation between
symmetric monoidal functors.
References
{{DEFAULTSORT:Monoidal Natural Transformation
Monoidal categories