Concerning the potential-functional embedding theory (PDFT), FDET expressions for the total energy and the embedding potential admit also the embedded wave-function and the environment density obtained from PDFT. In such a case, the total energies and embedding potentials of PFET and FDET are expected to be the same, in the absence of additional assumptions and approximations. In their exact form, both approaches target the same solutions. The identity of the corresponding quantities, one given as an explicit functional of the environment density (FDET) and one not (self-consistent PFET), might lead to a better understanding of the relevant density functionals. Concerning Cortona's formulation of DFT applied originally to atoms in solids, it is worthwhile to recall an intriguing observation regarding LDA made when it was used for intermolecular complexes in our exploratory works. For such complexes, LDA is known to be inadequate for approximating the exchange-correlation energy. However, when applied simultaneously for both the exchange-correlation and nonadditive kineticenergy functionals, LDA yields surprisingly good interaction energies. The reasons for this apparent compensation of errors remains an open question.