Commutative monoids have binary products

[imlala]

Homework Statement

Hello,

I'd like to prove that the category cmon of commutative monoids has binary products.

The Attempt at a Solution

actually I'm aware that i have to use cartesian products
given monoids (M, $$\bullet$$m, $$e^{}_{m}$$) and (N, $$\bullet$$n, $$e^{}_{n}$$)

it follows that (M$$\times$$N) $$\times$$ (M$$\times$$N) $$\rightarrow$$ M$$\times$$ N

and ((m,n), (m',n')) |---> (m $$\bullet$$m m', n $$\bullet$$n n') ...

Thanks in advance for any help!

 

[Hurkyl]

Well, what have you managed to do successfully, and where are you stuck? Can you, at least, write an outline of the individual steps you have to do even if you can't do them?


P.S. I'm assuming you don't have any general structure theorems available -- e.g. that any variety of universal algebra has products.

 

[imlala]

actually I've got the solution for the particular problem for category monoid of monoids but i can't figure out what is needed to add to the proof to satisfy the proposition that cmon of commutative monoids have binary products.