A comparison of twisted coefficient systems

In the first part of this note, we review and compare various instances of the notion of twisted coefficient system, a.k.a. polynomial functor, appearing in the literature. This notion hinges on how one defines the degree of a functor from $\mathcal{C}$ to an abelian category, for various different structures on $\mathcal{C}$. In the second part, we focus on twisted coefficient systems defined on partial braid categories, and explain a functorial framework for this setting.