Monoidal natural transformation

Template:Refimprove Suppose that $(𝒞,\otimes ,I)$ and $(𝒟,\bullet ,J)$ are two monoidal categories and

$(F,m):(𝒞,\otimes ,I)\to (𝒟,\bullet ,J)$ and $(G,n):(𝒞,\otimes ,I)\to (𝒟,\bullet ,J)$

are two lax monoidal functors between those categories.

A monoidal natural transformation

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

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

Template:Spaces and Template:Spaces

commute for every objects $A$ and $B$ of $𝒞$.

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

References

• Template:Cite book