  In arXiv:1506.05880 we gave a generalization of the theory of quivers with
potentials introduced by Derksen-Weyman-Zelevinsky, via completed tensor
algebras over $S$-bimodules where $S$ is a finite dimensional basic semisimple
algebra. In this paper we show how to extend this construction to the level of
decorated representations and we prove that mutation of decorated
representations is an involution. Moreover, we prove that there exists a nearly
Morita equivalence between the Jacobian algebras which are related via
mutation. This generalizes the construction given by Buan-Iyama-Reiten-Smith.
