We propose an effective density rho(pseudo) that engenders direct mapping of the free energy F-U of uniform fluids to the free energy F-N of nonuniform fluids by requiring F-N = F-U (at rho(pseudo)). The equality is called the congruence condition. It is made possible by considering the statistical mechanical theory: the potential distribution theorem (PDT). The PDT connects three quantities: the work W-ins(z) needed for inserting a test particle into the fluid, the chemical potential mu(0) of the bulk fluid, and the nonuniform singlet density A.,(1)(z). We use Monte Carlo (MC) data to obtain the insertion work W-ins(z) (via the Euler-Lagrange equation) from the probability densities rho((1))(w)(z). The concept of the congruent effective density rho(pseudo) is applicable to general interaction potentials, not restricted to the hard sphere type. We examine thus two simple fluids adsorbed on a hard wall: (i) the hard spheres and (ii) the Lennard-Jones fluid for comparison. We discern the difference in behavior of the effective density vis-a-vis whether there is enhancement or depletion of the fluid density near the wall (namely, if there is accumulation of molecules at the wall, or if there is deficit of molecules at the wall). rho(pseudo)(z) is found to exhibit for enhanced adsorption out-of-phase oscillations compared to rho((1))(w)(z). For depleted adsorption, we do not observe oscillations, and the trends of rho(pseudo)(z) are in line with those of rho((1))(w)(z). Explanation is sought in terms of the concavity of the chemical-potential function.