Jerry #154 - I think the "and..." part is necessary. For example:

Let \\(P=\\{a\\}\\), \\(Q=\\{1,2\\}\\). Let \\(f(a)=1\\) and \\(g(1)=g(2)=a\\). Then \\(f\\), \\(g\\) are monotone, \\(f.g=id_P\\) but \\(g.f\neq id_Q\\).

Let \\(P=\\{a\\}\\), \\(Q=\\{1,2\\}\\). Let \\(f(a)=1\\) and \\(g(1)=g(2)=a\\). Then \\(f\\), \\(g\\) are monotone, \\(f.g=id_P\\) but \\(g.f\neq id_Q\\).