For a finite dimensional algebra A over a field k the functor k-Alg\(\ni C\mapsto A\otimes C\in k\)-Alg has a left adjoint a(A,-). The construction of the algebra a(A,B) is dual to Sweedler’s construction of the universal measuring coalgebra and similar to Manin’s constructions on quadratic algebras. The universal property defines a bialgebra structure on a(A,A). A turns out to be an a(A,A)-comodule algebra. The author investigates a(A,B)-modules, their Ext groups, and a(A,A)-module algebras. He shows that the categories of a(A,A)-modules and of chain complexes of k-modules are equivalent as monoidal categories if \(\dim (A)>1\).