Monoidal functor

Concept in category theory

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 functorsThe 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.

This keyword could refer to multiple things. Here are some suggestions: