  This paper proves that homology equivalences of cogenerating complexes induce
homology equivalences of the cofree coalgebras in many interesting cases. We
show that the underlying chain complex of any cofree coalgebra is naturally a
direct summand of the underlying chain-complex of a cofree coalgebra over a
free operad. This is combined with the previous result to prove the homology
invariance of all cofree coalgebras.
