In this Note, we propose a new perspective on the foundations of the exact factorization (EF) representation of the molecular wavefunction, strengthening its connection to the more traditional Born-Huang (BH) representation, which invokes an expansion of the molecular wavefunction in terms of a basis of electronic states. Specifically, we demonstrate that EF is the representation of the molecular wavefunction in terms of eigenstates of the electronic density operator. From this viewpoint, we show that the U (1) gauge freedom in the definition of the electronic and nuclear amplitudes as well as the partial (electronic) normalization condition arise naturally when diagonalizing the density operator. Analogously, the EF evolution equations can be derived directly from BH instead of invoking the time-dependent variational principle applied to the molecular time-dependent Schrödinger equation (TDSE) ( Tn + Ĥel )Ψ = i∂ t Ψ, with Tn and Ĥel the nuclear kinetic and the electronic Hamiltonian operators, respectively.