Role of fiber functor monoidal structure in Tannakian bialgebra reconstruction Shahn Majid presents in section 9.4 of "Foundatations of Quantum Group Theory" a Tannaka-type reconstruction theorem for producing a $k$-bialgebra out of its $k$-linear monoidal category of modules $\mathcal{M}$. The theorem depends on the presense of a fiber (fogetful) functor $F: \mathcal{M} \rightarrow Vect_k$ and monoidal structure on $F$. The question is:

Is there a stronger reconstruction theorem in which $F$ carries no monoidal structure?

If not, it means there should be an example with different monoidal structures on $F$ yielding different coproducts in the bialgebra. Is there such an example?
 A: Etingof and Gelaki (and others) constructed examples of isocategorical groups, that is non-isomorphic finite groups $G$ and $H$ such that the representations categories $Rep(G)$ and
$Rep(H)$ (over the complex numbers) are equivalent as monoidal categories. Now if you use the usual fiber functor $Rep(G)\to Vect$ you will reconstruct the group algebra of $G$ (or its dual). And if you use the composition $Rep(G)\simeq Rep(H)\to Vect$ you will reconstruct the group algebra of $H$. 
A: The most basic form of Tannaka duality goes as follows:
Given a $k$-coalgebra $C$, we can construct its category of finite dimensional comodules ("representations"), which carries a (faithful, exact) functor to the category of finite dimensional vector spaces. On the other hand, given a $k$-linear abelian category equipped with a functor (the "fiber functor") to the category of finite dimensional vector spaces, we can construct a (not necessarily finite dimensional) $k$-coalgebra using a certain coend formula. Tannaka duality says that these two functors give an adjunction between the category of $k$-coalgebras and the 2-category of $k$-linear abelian categories equipped with a fiber functor. The counit of this adjunction is an isomorphism (so that we can reconstruct the coalgebra from its category of representations), and the unit of this adjunction is an equivalence precisely when we restrict to categories for which the fiber functor is faithful and exact (so that we can recognize such categories and fiber functors as being the ones that are categories of representations).
Note that the above result is entirely trivial (i.e., follows from the Yoneda lemma) if we take the category of not-necessarily finite dimensional comodules or require that our coalgebras be finite dimensional. On the other hand, the above result is entirely false if we replace coalgebras and comodules with algebras and modules. (Exercise: Find a nontrivial algebra that has no nontrivial finite dimensional modules. It's pretty easy!)
The adjunction I described can be "souped up" to a monoidal adjunction to handle representations of bialgebras; the algebra structure corresponds to the monoidal structure on the category. But the same caveats apply, so the theorem you state about monoidal categories is either trivial or wrong for the above reasons. The "correct" statement of monoidal Tannaka duality (and I believe more or less the one stated by Deligne) involves on the one side commutative Hopf algebras and on the other side symmetric monoidal $k$-linear abelian categories equipped with a faithful, exact, symmetric monoidal functor to finite-dimensional vector spaces. This is just a symmetric monoidal-ification of the basic form I give above.
A good reference is this paper by Daniel Schäppi (particularly the opening section), and its successor.
A: To give a different answer to your meta-question, you can certainly have a non-monoidal fiber functor from C -> Vec where C is not the category of representations of any bialgebra.  Just take the category of twisted G-graded vector space Vec(G,w) where w is a nontrivial 3-cocycle.  This is not the category of representations of any bialgebra, but it has an obvious non-monoidal fiber functor to vector spaces.  (It is, however, the category of representations of a quasi-bialgebra.)
A: The answer to your question is no. Take a finite set S with two non-isomorphic monoid structures (for example, the four element set with its two group structures). Let $k^{(S)}$ be the algebra of $k$-valued functions on $S$ with pointwise multiplication. Both monoid structures on $S$ induce a comultiplication on $k^{(S)}$. Hence they define two monoidal structures on the category of $k^{(S)}$-modules, together with strong monoidal structures on the forgetful functor $F$ to vector spaces. Thus, if you forget about the monoidal structure of $F$ you can reconstruct $k^{(S)}$ as an algebra, but you won't be able to reconstruct the comultiplication.
