We consider a nucleating center consisting of both an insoluble core and soluble species and develop the thermodynamics of isothermal formation of a droplet on such a nucleus in a multicomponent vapor mixture. Two different approaches to the derivation of the free energy of droplet formation within the framework of the capillarity approximation are considered. If condensation is not barrierless, the free energy of formation describes a multidimensional free-energy surface having a "well" point and a "saddle" point. It is shown that in a strict theory, taking account of surface enrichment effects, the compositions of droplets corresponding to these two points are equal and can be found without knowing the surface tension of the droplet. For the case of no surfactants in the droplet, we extend the Kuni method of investigating the behavior of the free energy of droplet formation to the case of heterogeneous multicomponent condensation on mixed nuclei, which makes it possible to find out all the main features of the free-energy surface without explicitly knowing the free-energy itself. The theoretical results are illustrated by numerical calculations for the water-methanol condensation on mixed nuclei.