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.)

Let me clarify my question. For a commutative algebra $A$ we define a functor $F:\mathrm{Set}_\mathrm{f}\to\mathrm{Vect}$ by

$F(I)=A^{\otimes I}$ for a finite set $I$ and

$F(t):F(I)\to F(J)$, $\bigotimes_{i\in I}a_i\mapsto\bigotimes_{j\in J}\prod_{i\in t^{-1}(j)}a_i$ for a map $t:I\to J$ (exactly as Andreas Blass proposed).

Suppose now that two finite-dimensional algebras $A$ and $B$ over the complex numbers produce isomorphic functors $F$ and $G$. Is it true that then $A$ and $B$ are isomorphic?

The question is not trivial. Let $e:F\to G$ be an isomorphism of functors. Then $e_{\{1\}}:A\to B$ and $e_{\{1,2\}}:A\otimes A\to B\otimes B$ are isomorphisms of vector spaces. If we had $e_{\{1,2\}}=e_{\{1\}}\otimes e_{\{1\}}$, this would imply that $e_{\{1\}}$ is an isomorphism of algebras. The problem is that we have only linear naturality relations between $e_I$.