First Hohenberg-Kohn theorem says the ground-state energy from the Schorödinger equation is a unique functional of the Electron density.
There is a one-to-one correspondence between the ground state density of a many-electron system and the external potential.
The external potential (and hence the total energy), is a unique functional of the electron density.
In the previous descriptions, “a unique functional of …” is the same as “a one-to-one correspondence between”. There are many ways to describe energy including ground-state energy, and external potential energy.