Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X + Y + X \times Y$ and unit given by $I := 0$.
This monoidal structure is alluded to in several places online (e.g. this thread on the categories mailing list), but I'm having more trouble finding an explicit reference in the literature (including a proof that this does indeed define a monoidal category).
Where is this example treated? The more generality the better (e.g. it would be ideal to have a reference for the rig category case), but I would be satisfied with the case of distributive categories. (I don't even know an explicit reference for $\mathscr C = \mathbf{Set}$.) I would be even happier to have a reference in which the monoids are also characterised (I believe they ought to be the semigroups in $\mathscr C$.)
