It's also interesting to note that for \\(f(a) = 2a\\), \\(R_f(b) \leq L_f(b) \; \forall b\\) but for the vector space example, when a left adjoint \\(L_T\\) is defined, \\(L_T(w) \subseteq R_T(w) \; \forall w \subseteq W\\).

This is why I find that the "cautious versus generous" phrasing has not helped me develop my intuition here.

Note that by taking the cross product of the two partial orders, we can construct a partial order for which the left and right adjoints do not have either of the two relations \\(L \leq R\\) or \\(R \leq L\\) consistently across their whole range.