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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 * The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible. * The coherence maps of strong monoidal functors are invertible. * The coherence maps of strict monoidal functors are identity maps. Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors. == Definition == Let and be monoidal categories. A monoidal functor from to consists of a functor together with a natural transformation : between functors and a morphism :, called the coherence maps or structure morphisms, which are such that for every three objects , and of the diagrams :, : and commute in the category . Above, the various natural transformations denoted using are parts of the monoidal structure on and . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「monoidal functor」の詳細全文を読む スポンサード リンク
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