Suppose that

(\mathcal C,\otimes,I)

and

(\mathcal D,\bullet, J)

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 ...

and :

(F,m):(\mathcal C,\otimes,I)\to(\mathcal D,\bullet, J)

and

(G,n):(\mathcal C,\otimes,I)\to(\mathcal D,\bullet, J)

are two

lax monoidal functor In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with tw ...

s between those categories. A monoidal natural transformation :

\theta:(F,m) \to (G,n)

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 na ...

\theta:F \to G

between the underlying functors such that the diagrams : and commute for every objects

A

and

B

\mathcal C

(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