The notation is supposed to be suggestive of a product with an object . While this is the motivating example, the interval functor need not be of that form.
Cylindrical model structures
Richard Williamson has developed a way to build a model structure from the simple point of departure of a structured interval in a monoidal category - more generally, a structured cylinder and a structured co-cylinder in a category. This is given in his thesis and also in the ArXiv article listed below.