This seminar focuses on describing two different algebraic models of configuration spaces: the simplicial Barratt-Eccles model, equipped with a coalgebra structure induced by the standard cup product for simplicial sets, and the multisimplicial surjection model, endowed with a coalgebra structure as a natural extension of the simplicial case. The main goal is to investigate the connection between these two descriptions, demonstrating the compatibility of their respective coalgebra structures. We will outline an explicit map between them, from a purely combinatorial perspective.