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

• 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.