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 and be monoidal categories. A lax monoidal functor from to (which may also just be called a monoidal functor) consists of a functor together with a natural transformation

from functors to and a morphism


called the coherence maps or structure morphisms, which are such that for every three objects , and of the diagrams


commute in the category . Above, the various natural transformations denoted using are parts of the monoidal structure on and .



Alternate notions

If and are closed monoidal categories with internal hom-functors (we drop the subscripts for readability), there is an alternative formulation

ψAB : F(AB) → FAFB

of φAB commonly used in functional programming. The relation between ψAB and φAB is illustrated in the following commutative diagrams:

Commutative diagram demonstrating how a monoidal coherence map gives rise to its applicative formulation
Commutative diagram demonstrating how a monoidal coherence map can be recovered from its applicative formulation


Monoidal functors and adjunctions

Suppose that a functor is left adjoint to a monoidal . Then has a comonoidal structure induced by , defined by



If the induced structure on 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.

See also