Restriction of Scalars for $L_{\mathrm{\infty }}$-Modules

Let $I:L^{\mathrm{\prime }}\to L$ be a morphism of $L_{\mathrm{\infty }}$-algebras. The goal of this paper is to describe restriction of scalars in the setting of $L_{\mathrm{\infty }}$-modules and prove that it defines a functor $I^{\ast }:L\text{-mod}\to L^{\mathrm{\prime }}\text{-mod}$. 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 $L_{\mathrm{\infty }}$-module structure on the sutured annular Khovanov homology of a link in a thickened annulus.