With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
A monoidal functor is a functor between monoidal categories that preserves the monoidal structure: a homomorphism of monoidal categories.
Let and be two monoidal categories. A lax monoidal functor between them is
a natural transformation
satisfying the following conditions:
(associativity) For all objects the following diagram commutes
where and denote the associators of the monoidal categories;
(unitality) For all the following diagrams commutes
where , , , denote the left and right unitors of the two monoidal categories, respectively.
If and all are isomorphisms, then is called a strong monoidal functor. (Note that ‘strong’ is also sometimes applied to ‘monoidal functor’ to indicate possession of a tensorial strength.)
Lax monoidal functors send monoids to monoids.
If is a lax monoidal functor and
is a monoid in , then the object is naturally equipped with the structure of a monoid in by setting
This construction defines a functor
between the categories of monoids.
Similarly, an oplax monoidal functor sends comonoids to comonoids.
Just like monoidal categories, monoidal functors have a string diagram calculus; see these slides for some examples.