Restriction of Scalars for $L_\infty$-Modules

Let $I: L' \to L$ be a morphism of $L_\infty$-algebras. The goal of this paper is to describe restriction of scalars in the setting of $L_\infty$-modules and prove that it defines a functor $I^*: L\text{-mod} \to L'\text{-mod}$. A more abstract approach to this problem was recently given by Kraft-Schnitzer. In a subsequent paper, this result is applied to show that there is a well-defined $L_\infty$-module structure on the sutured annular Khovanov homology of a link in a thickened annulus.