I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people here at MO are more active, so I hope, specialists in the category theory could clarify the remaining part of my question. (At the same time I foresee that somebody can suggest another idea of how this can be proved, and I would welcome this as well.)

The nLab article on the Tannaka duality says that this theory was generalized to monoids in arbitrary closed monoidal category (symmetric and complete in some sense):

if we take a monoid $A$ in such a category $\mathcal V$ and consider the category $_A{\mathcal V}$ of all left modules over $A$, then $A$ can be recovered from $_A{\mathcal V}$ as the object of enriched endomorphisms of the forgetful functor $F: {_A{\mathcal V}}\to {\mathcal V}$.

(As far as I understand, this result is called a reconstruction theorem.)

Some details in this construction are however explained at nLab vaguely, in particular,

how is the structure of the enriched category introduced on $_A{\mathcal V}$?

Can anybody give me a link to a text where this result is explained accurately so that I could refer to it with a clear conscience? (Perhaps I am missing something, I do not see where this is written. From what I see, I have the impression that this is folklore, but I have no confidence, and I do not dare to write this in my paper.)