  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.
