I don't remember how to define nature in the book, but from the two examples you cited, we can see that the scope of application of the same increase and different decrease is that the outer function is monotonous, and the inner function can be piecewise monotonous, and vice versa.
The positive application can explain x2>x 1, u2>u 1, y2>Y 1 (increase), and other situations can also be directly proved.
It is not difficult to understand that the reverse is not applicable. The independent variable of the external function y=f(u) is u, not x.
Suppose that u=g(x) increases monotonically, for x2 >: X 1, U2 >: U 1, but in which segment of y=f(u) U2 and U 1 are uncertain, so the size relationship cannot be obtained.
Taking the above topic as an example, the dividing point of f(u) is u= 1/2, which corresponds to x= 1, not x= 1/2.