Take some element in the dual cone $g\in L^1([0,1],Y_+)^*\subset L^\infty([0,1],Y^*)$ then by definition for every $f\in L^1([0,1],Y_+)$
$$\int_0^1 <f(t),g(t)>\mathbb{d}t\geq0$$
In particular for any $y\in Y_+$ and $[a,b]\subset [0,1]$ take $f \equiv y\cdot \chi_{[a,b]}$ then
$$\int_a^b <y,g(t)>\mathbb{d}t\geq0$$
hence $<y,g(t)>\geq 0$ for every $t\in[0,1],y\in Y_+$ thus the image of $g$ is in the positive cone which means $$g\in L^\infty([0,1],Y_+^*)$$