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 two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors.

Let ${\displaystyle ({\mathcal {C}},\otimes ,I_{\mathcal {C}})}$ and ${\displaystyle ({\mathcal {D}},\bullet ,I_{\mathcal {D}})}$ be monoidal categories. A monoidal functor from ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle {\mathcal {D}}}$ consists of a functor ${\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}$ together with a natural transformation

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