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.