See also: strict monoidal multicategory, where I detail the equations.
Remark. Multifunctors between representable multicategories are lax monoidal functors, and multinatural transformations are lax monoidal transformations. This implies that a representable monoidal multicategory is exactly a duoidal category.
See also.
References
- nLab, Mike Shulman: Virtual duoidal category.