Local systems with restricted variation on the formal punctured disc via factorization

We define the stack of $G$-local systems with restricted variation on the formal puntured disc and study its properties. We embed sheaves of categories over this stack into the category of factorization module categories over $\operatorname{Rep}(G)$. Along the way we develop a theory of factorization structures in families and study functorialities of such under changes of the base curve.