A commutative algebra (with unity) over a field gives rise to the covariant functor F: Set_f->Vect from finite sets to vector spaces: F(E) := A^{otimes E}. Is it true that, over complex numbers, a finite dimensional algebra can be reconstructed from the corresponding functor?

(A Gamma-module is a functor from finite pointed sets to vector spaces; so F is not a Gamma-module. I use this term in the title just because I do not know the correct term for F: Set_f->Vect.)