Let us first recall that a dendriform algebra [8, 9, 10, 5] is a family (A, [??], [??]) such that A is a vector space and [??], [??] are two products on A, satisfying three axioms.
Definition 4.1 (Definition 2 of ) A dendriform coalgebra is a family (C, [[DELTA].sub.[??]], [[DELTA].sub.[??]]) such that:
Proposition 4.2 The triplet ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a dendriform coalgebra.
Indeed, the necessary compatibilities for the dendriform coalgebra ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) to be a codendriform bialgebra  write:
It is itself a subalgebra of WQSym, based on packed words (or set compositions), in which the role of PBT is played by the free dendriform trialgebra on one generator TD (based on Schroder trees), the free cubical trialgebra TC (segmented compositions).
The realization of the free dendriform trialgebra given in  involves the following construction.
CHAPOTON, Some dendriform functors, arXiv:0909.2751.