Functorial homotopy decompositions of looped co-H spaces

In recent work of the first and third authors, functorial coalgebra decompositions
of tensor algebras were geometrically realized to give functorial homotopy decompositions of loop suspensions. Later work by all three authors generalized this to functorial decompositions of looped coassociative co-H spaces. In this paper we use different methods which allow for the coassociative hypothesis to be removed.