In "Homotopy invariant algebraic structures on topological spaces" by Boardman and Vogt [BV], the authors characterize Stasheff's $A_{\infty}$-maps (for $A_{\infty}$-spaces without unit) using its M-construction (page 20, Definition 1.23 [BV]).
Given a map f from a $A_\infty$-space (WU-space in [BV]) X to a monoid Y, I do not see how the M-construction encodes de monoid structure of Y. It seems that it considers just the X structure... I do not see how the M-construction gives the compatibility between f and the monoid multiplication (see pages 9 and 10, Definition 1.13, case l=r).
I thank any hint.