Monoidal natural transformation
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Suppose that ( C , ⊗ , I ) {\displaystyle ({\mathcal {C}},\otimes ,I)} and ( D , ∙ , J ) {\displaystyle ({\mathcal {D}},\bullet ,J)} are two monoidal categories and
( F , m ) : ( C , ⊗ , I ) → ( D , ∙ , J ) {\displaystyle (F,m):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)} and ( G , n ) : ( C , ⊗ , I ) → ( D , ∙ , J ) {\displaystyle (G,n):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)}
are two lax monoidal functors between those categories.
A monoidal natural transformation
θ : ( F , m ) → ( G , n ) {\displaystyle \theta :(F,m)\to (G,n)}
between those functors is a natural transformation θ : F → G {\displaystyle \theta :F\to G} between the underlying functors such that the diagrams
and
commute for every objects A {\displaystyle A} and B {\displaystyle B} of C {\displaystyle {\mathcal {C}}}.
A symmetric monoidal natural transformation is a monoidal natural transformation between symmetric monoidal functors.