Graphical composition of mapping spaces between modules of configuration-space-type

In embedding calculus, spaces of embeddings are identified with derived mapping spaces between framed Fulton-MacPherson-type modules (framed configuration spaces). Unfortunately, there are no sufficiently good algebraic models for framed Fulton-MacPherson modules that would allow us to explicitly describe the rational homotopy type of the embedding space. Recently there were several attempts to avoid dealing with the framed versions of Fulton-MacPherson modules by considering framed manifolds, e.g. embeddings modulo immersions $\overline{\mathrm{Emb}}$ in a recent paper by Fresse, Turchin and Willwacher, or embeddings with a deformation of the framing $\widetilde{\mathrm{Emb}}$ in a recent paper by the author. In both cases, the rational homotopy type of the corresponding embedding space has an explicit description in terms of graphs (hairy graph complexes). We construct a combinatorial graphical composition for the composition of embedding spaces $\widetilde{\mathrm{Emb}}$. As a step on our way, we describe the action of the coinduction functor on configuration-space-type $e_n^c$-comodules.