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**.

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

- ,

- and

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

- The dual of a monoidal functor is a
**comonoidal functor**; it is a monoidal functor whose coherence maps are reversed. Comonoidal functors may also be called opmonoidal, colax monoidal, or oplax monoidal functors. - A
**strong monoidal functor**is a monoidal functor whose coherence maps are invertible. - A
**strict monoidal functor**is a monoidal functor whose coherence maps are identities. - A
**braided monoidal functor**is a monoidal functor between braided monoidal categories (with braidings denoted ) such that the following diagram commutes for every pair of objects*A*,*B*in :

- A
**symmetric monoidal functor**is a braided monoidal functor whose domain and codomain are symmetric monoidal categories.

- The underlying functor from the category of abelian groups to the category of sets. In this case, the map sends (a, b) to ; the map sends to 1.
- If is a (commutative) ring, then the free functor extends to a strongly monoidal functor (and also if is commutative).
- If is a homomorphism of commutative rings, then the restriction functor is monoidal and the induction functor is strongly monoidal.
- An important example of a symmetric monoidal functor is the mathematical model of topological quantum field theory, which has been recently developed. Let be the category of cobordisms of
*n-1,n*-dimensional manifolds with tensor product given by disjoint union, and unit the empty manifold. A topological quantum field theory in dimension*n*is a symmetric monoidal functor - The homology functor is monoidal as via the map .

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

*ψ*_{AB}:*F*(*A*⇒*B*) →*FA*⇒*FB*

of *φ*_{AB} commonly used in functional programming. The relation between *ψ*_{AB} and *φ*_{AB} is illustrated in the following commutative diagrams:

- If is a monoid object in , then is a monoid object in .

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

and

- .

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.