The logical structure and basic theorems of time-dependent current density functional theory (TDCDFT) are analyzed and reconsidered from the point of view of recently proposed time-dependent deformation functional theory (TDDefFT). It is shown that the formalism of TDDefFT allows to avoid a traditional external potential-to-density/current mapping. Instead the theory is formulated in a form similar to the constrained search procedure in the ground state DFT. Within this formulation of TDCDFT all basic functionals appear from the solution of a constrained universal many-body problem in a comoving reference frame, which is equivalent to finding a conditional extremum of a certain universal action functional. As a result the physical origin of the universal functionals entering the theory, as well as their proper causal structure become obvious. In particular, this leaves no room for any doubt concerning the predictive power of the theory.
A rigorous formulation of time-dependent current density functional theory (TDCDFT) on a lattice is presented. The density-to-potential mapping and the
V-representability problems are reduced to a solution of a certain nonlinear lattice Schr\"odinger equation, to which the standard existence and uniqueness results for nonlinear differential equations are applicable. For two versions of the lattice TDCDFT, we prove that any continuous-in-time current density is locally
V-representable (both interacting and noninteracting), provided that in the initial state the local kinetic energy is nonzero everywhere. In most cases of physical interest, the
V-representability should also hold globally in time. These results put the application of TDCDFT to any lattice model on a firm foundation, and pave the way for studying exact properties of exchange-correlation potentials.