Jonathan, you saved me, thanks a lot!

I completely forgot about associativity requirement!

**EDIT:** as with the monoidal category with 2 morphisms, the one with 3 gives rise to only 1 group and 6 monoids, so 6 rings may be formed from these.

The group (up to isomorphism) looks like \\(s_2 \circ s_2 = s_3\\), \\(s_3 \circ s_3 = s_2\\) and \\(s_2 \circ s_3 = s_3 \circ s_2 = 1\\) and is abelian - it's \\(\mathbb{Z}/3\\).