Free Monoids in Closed Symmetric Monoidal Categories There appear to be questions perhaps tangentially related to this that have been asked already.  If so a reference and a close would be heartily appreciated.
Give some category $\mathcal{C}$ with the modifiers listed in the title of the question, and an object $A$ in that category, how does one define the "free monoid" on that object?  Is this related to being an algebra over some monad?  
Thanks!
 A: Very simply, if a closed symmetric monoidal category has countable coproducts, and if monoid means monoid with respect to the monoidal product, then the free monoid on an object $A$ can be constructed as the "geometric series" 
$$F(A) = \sum_{n \geq 0} A^{\otimes n}.$$ 
The key fact needed to prove this is that $\otimes$ distributes over countable coproducts, and this is guaranteed by the closedness (indeed, $X \otimes -$ preserves arbitrary colimits). The proof that this is correct must be in thousands of places; see for instance Categories for the Working Mathematician (2nd edition), page 172. 
A: The 2nd edition of Mac Lane's Categories for the Working Mathematician has a description/construction of the free monoid on a given object in a monoidal category satisfying some side conditions, see Section 7.3. I am not sure off the top of my head if those conditions are met for every symmetric closed monoidal category, but they certainly hold for Set, K-Mod, Ban${}_1$ and other familiar examples.
(I don't think you can get free monoidal objects in Ban, but I have never written down exact details, and suspect that's not really an example of direct interest to you.)
