Question on (co)monoids

[claudio pisani <pisclau@yahoo.it>]

Dear categorists,

in several places I have seen variants of the following statement (or its dual):

If C is a symmetrical monoidal category and every object has a natural monoid structure (that is any map is a monoid  morphism) then C is cocartesian monoidal (tensor = sums).

I have not found any proof of the general statement. 
In fact the tensor product of two objects has obvious maps A -> AxB <-B and for any A -> X <- B it's easy to see that there is at least one factorization AxB -> X.
The problem is unicity. If the monoid structure on objects is unique, then that on AxB (and on I) has to be the usual one and unicity follows.

Question: is the statement true in the above general form or one has to assume that the monoid structure on objects is unique? Or that it is commutative?

I would be grateful for any suggestion or reference.

Claudio




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