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.

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.