I explain how a category graded by a monoidal category can be viewed as a double functor out of the delooping of the monoidal category. A few consequences and a series of examples are then presented.
A very important class of examples are the categories of parameterized morphisms, that is taking \mathcal{C}_m(x,y) = \operatorname{Hom}(m \cdot x, y) for some action of the monoidal category \mathcal{M}. In fact I think this should give a 2-equivalence of category actions with a subcategory of graded categories (the “representable” ones), but I’m not aware of this being written down anywhere.
While it’s not stated in exactly this language, this perspective on actions of a monoidal category appears in Wood’s thesis on graded categories (see pages 41 – 43 of Wood’s “Indicial methods for relative categories”). For a statement in more modern language, see for instance Proposition 4.14 of Campbell’s “Skew-enriched categories”.
Thanks both for the comments! Eigil, also have a look at Example 3.9 in Lucyshyn-Wright’s paper, where V-actegories are shown to be 2-equivalent to the full sub-2-category of V-graded categories spanned by the V-graded categories with V-copowers. Lucyshyn-Wright points out that this goes back to Wood, as Nathanael says.