Gray monoid
A Gray monoid is a Gray category with a single object. They are a semistrict version of monoidal bicategories.
Gray monoids are 2-categories with associative and unital monoidal product $(\otimes)$, which however fails to be a 2-functor (see Monoidal Bicategories and Hopf Algebroids (Day, Street, 1997)). When working with monoidal bicategories, we can safely assume we are working with Gray monoids, as there is a coherence result.