Late to the party here, and actually stuck with a simple question, maybe someone will return back and will help to resolve my struggle :)

As Johnatan shown in his [comment](, the monoidal category with 2 morphisms, \\(1\\) and \\(f\\) has only 2 "realizations", so there are only two possible such categories (\\(f \circ f = f\\) and \\(f \circ f = 1\\)).

And, according to OEIS, there are 7 possible monoidal categories with 3 morphisms. Following the logic, if we have 3 morphisms \\(1, s_1, s_2\\), then \\(s_1 \circ s_2 \\) may be any of these 3, similar reasoning goes for \\(s_1 \circ s_1\\),
\\(s_2 \circ s_2\\), and \\(s_2 \circ s_1\\). So if we are not counting up to isomorphism, there are \\(3^4=81\\) such categories. And then I've tried to reduce this number to 7 with no luck, even taking into account 2 permutations of \\(s_1\\) and \\(s_2\\), the number is not even close.

So how one achieves this magic number 7? :)