On categories with arbitrary 2-cell structures

When a category is equipped with a 2-cell structure it becomes a sesquicategory but not necessarily a 2-category. It is widely accepted that the latter property is equivalent to the middle interchange law. However, little attention has been given to the study of the category of all 2-cell structures (seen as sesquicategories with a fixed underlying base category) other than as a generalization for 2-categories. The purpose of this work is to highlight the significance of such a study, which can prove valuable in identifying intrinsic features pertaining to the base category. These ideas are expanded upon through the guiding example of the category of monoids. Specifically, when a monoid is viewed as a one-object category, its 2-cell structures resemble semibimodules.