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
between formula_7 functors and a morphism
called the coherence maps or structure morphisms, which are such that for every three objects formula_9, formula_10 and formula_11 of formula_3 the diagrams
commute in the category formula_4. Above, the various natural transformations denoted using formula_14 are parts of the monoidal structure on formula_3 and formula_4.
Monoidal functors and adjunctions.
Suppose that a functor formula_5 is left adjoint to a monoidal formula_42. Then formula_43 has a comonoidal structure formula_44 induced by formula_45, defined by
and
If the induced structure on formula_43 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.