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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]