Monoidal natural transformation

Monoidal natural transformation

Suppose that (\mathcal C,\otimes,I) and (\mathcal D,\bullet, J) are two monoidal categories 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 functors between those categories.

A monoidal natural transformation

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

between those functors is a natural transformation \theta:F\Rightarrow G between the underlying functors such that the diagrams

Monoidal nat transfo mult.png and Monoidal nat transfo unit.png

commute for every objects A and B of \mathcal C.

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


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