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. Definition. Let formula_1 and formula_2 be monoidal categories. A lax monoidal functor from formula_3 to formula_4 (which may also just be called a monoidal functor) consists of a functor formula_5 together with a natural transformation from functors formula_7 to formula_8 and a morphism called the coherence maps or structure morphisms, which are such that for every three objects formula_10, formula_11 and formula_12 of formula_3 the diagrams commute in the category formula_4. Above, the various natural transformations denoted using formula_15 are parts of the monoidal structure on formula_3 and formula_4. Alternate notions. If formula_1 and formula_2 are closed monoidal  categories with internal hom-functors formula_40 (we drop the subscripts for readability), there is an alternative formulation of "φ" commonly used in functional programming. The relation between "ψ" and "φ" is illustrated in the following commutative diagrams: Monoidal functors and adjunctions. Suppose that a functor formula_5 is left adjoint to a monoidal formula_46. Then formula_8 has a comonoidal structure formula_48 induced by formula_49, defined by and If the induced structure on formula_8 is strong, then the unit and counit of the adjunction are monoidal natural transformations, and the adjunction is said to be a monoidal adjunction; conversely, the left adjoint  of a monoidal adjunction is always a strong monoidal functor. Similarly, a right adjoint to a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is a strong monoidal functor.