Keywords General-covariant multi-component systems · Entropy Identity · Entropy balance of a component of the mixture · Entropy balance of the mixture · Multi-temperature relaxation · Equilibrium conditions: 4-temperature's Killing relation · Extended Belinfante/Rosenfeld procedure · 2-component plain-ghost mixture Abstract Non-equilibrium and equilibrium thermodynamics of an interacting component in a relativistic multi-component system is discussed covariantly by exploiting an entropy identity. The special case of the corresponding free component is considered. Equilibrium conditions and especially the multi-component Killing relation of the 4-temperature are discussed. Two axioms characterize the mixture: additivity of the energy momentum tensors and additivity of the 4-entropies of the components generating those of the mixture. The resulting quantities of a single component and of the mixture as a whole, energy, energy flux, momentum flux, stress tensor, entropy, entropy flux, supply and production are derived. Finally, a general relativistic 2-component mixture is discussed with respect to their gravitation generating energy-momentum tensors. member of the mixture is investigated. Thus, each component of the mixture is equipped with its own temperature, pressure, energy and mass density which all together generate the corresponding quantities of the mixture.Considering a multi-component system, three items have to be distinguished: one component as a member of the multi-component system which interacts with all the other components of the system, the same component as a free 1-component system separated from the multi-component system and finally the multi-component system itself as a mixture which is composed of its components. Here, all three items are discussed in a covariant-relativistic framework. For finding out the entropy-flux, -supply, -production and -density, a special tool is used: the entropy identity which constrains the possibility of an arbitrary choice of these quantities [9,10,11,12]. Following J. Meixner and J.U. Keller that entropy in non-equilibrium cannot be defined unequivocally [13,14,15,16,17], the entropy identity is only an (well set up) ansatz for constructing a non-equilibrium entropy and further corresponding quantities. This fact in mind, a specific entropy and the corresponding Gibbs and Gibbs-Duhem equations are derived. The definition of the rest mass flux densities, of the energy and momentum balances and of the corresponding balances of the spin tensor are taken into account as contraints in the entropy identity by introducing fields of Lagrange multipliers. The physical dimensions of these factors allow to determine their physical meaning.Equilibrium is defined by equilibrium conditions which are divided into basic ones given by vanishing entropy-flux, -supply and -production and into supplementary ones such as vanishing diffusion flux, vanishing heat flux and zero rest mass production [11,12]. The Killing relation of the 4-temperature concerning equilibrium is shortly d...