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 $$g\in L^\infty([0,1],Y_+^*)$$

Take some element in the dual cone $g\in L^1([0,1],Y_+)^*\subset L^\infty_{w^*}([0,1],Y^*)$. Then, by definition, for every $f\in L^1([0,1],Y_+)$

$$\int_0^1 \langle f(t),g(t)\rangle\mathbb{d}t\geq0$$

In particular, for any $y\in Y_+$ and $[a,b]\subset [0,1]$, we can take $f \equiv y\cdot \chi_{[a,b]}$ so that

$$\int_a^b \langle y,g(t)\rangle \mathbb{d}t\geq 0.$$

Hence, for every $y\in Y_+$, and every $t\in[0,1]$ we have $\langle y,g(t) \rangle\geq 0$. Thus the image of $g$ is in the positive cone which means that $$g\in L^\infty([0,1],Y_+^*).$$