Tensor products in concrete categories

In this paper we consider the notion of tensor multiplication in the concrete categories (by the concrete category we mean the category with fixed covariant faithful functor U : -- Ens). The reason of this choice is the observation of the constructions of tensor product in the categories of abelian groups, vector spaces or more generally in any variety (which are of course concrete). We modify this constructions to give the universal method of introduction the tensor multiplication in any concrete category. Moreover we are not restricted because many important categories are concrete. Our aim was the general overview on the tensor multiplication in order to apply it to objects in any category which fulfill suficient conditions. In order to do this we use the construction of tensor product via Freyd's representability theorern ([4], [1]). This allowed us to formulate the probIem in the language of theory of category. The main result of this work is theorem 2 which gives the conditions sufficient to existence the tensor product in the concrete category. As an exarnple of the nontrivial aplication of this theorem we give the proof of the existence of the tensor product in the category of compact spaces.