In this paper we introduce the notion of generalized factorization for an arbitrary submonoid \(M\subseteq A^*\) , where \(A^*\) is the free monoid generated by an alphabet A, generalizing, in this way, the notion of factorization of \(A^*\) . Then we give a characterization of the free product of two submonoids of \(A^*\) in terms of unambiguous products of monoids. To do this we make use of the notion of coding partition of a set \(X\subseteq A^+\) , where \(A^+\) is the free semigroup generated by an alphabet A. Moreover, given a coding partition of a set \(X\subseteq A^+\) , we will show how to construct a generalized factorization of \(X^*\) .