  We apply our previous work on cluster characters for Hom-infinite cluster
categories to the theory of cluster algebras. We give a new proof of
Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster
algebras IV for skew-symmetric cluster algebras. We also construct an explicit
bijection sending certain objects of the cluster category to the decorated
representations of Derksen, Weyman and Zelevinsky, and show that it is
compatible with mutations in both settings. Using this map, we give a
categorical interpretation of the E-invariant and show that an arbitrary
decorated representation with vanishing E-invariant is characterized by its
g-vector. Finally, we obtain a substitution formula for cluster characters of
not necessarily rigid objects.
