duoidal category of graded objects
The original version by Vasilakopoulou and López Franco uses the coproduct structure, preserved by the tensor.
Technical remark. These categories can be used to define graded monoids and graded comonoids as monoids and comonoids. However, one needs to be careful of picking the correctly op’d category. Additionally, it seems that graded comonoids can be defined in a greater generality: we would be fine with grading a promonad and having a produoidal structure – this would avoid the strong condition of the terminal preserving the tensor.
References.