Mutation of representations and nearly Morita equivalence

Buan-Iyama-Reiten-Smith proved, based on Derksen-Weyman-Zelevinsky work, that the Jacobian algebra of two quivers with potential related by a QP-mutation are nearly Morita equivalent. They proved, using Axiom of Choice, that the natural functor $\mu_k$ is an equivalence by showing that $\mu_k$ is full, faithfull and dense. In this note we provide a quasi-inverse $\mu_k^-$ to $\mu_k$ without Axiom of Choice.