Suppose I have an irreducible representation $\phi:\mathfrak{sl}_3 \to \mathfrak{gl}(V)$ of the Lie algebra $\mathfrak{sl}_3$. Now I have been asked to express the heighest weight of the corresponding dual representation on $\mathfrak{gl}(V^\ast)$ in terms of those for that on $V$. The definition I have of a weight is: A linear function $\mu : \mathfrak{h} \to \Bbb{C}$ is said to be a weight for $\phi$ if there is $v \in V$ such that
$\phi(H)v = \mu(H)v$
for all $H \in \mathfrak{h}$. $\mathfrak{h}$ is the usual Cartan subalgebra of $\mathfrak{sl}_3$. Now I seem to only be able to calculate explicitly the highest weight of $V^\ast$ in the case that I have a concrete representation, such as the standard representation. Furthermore, what does one mean by "express the highest weight of $V^\ast$" in terms of that for $V$?
For example the highest weight of the standard representation of $\mathfrak{sl}_3$ is the linear functional $L_1$ defined by
$L_i \left( diag(a_1,a_2,a_3) \right) = a_i \hspace{2mm} \text{for $i=1,2,3$}.$
Here $diag(a_1,a_2,a_3)$ is a matrix in the Cartan subalgebra $\mathfrak{h}$. The highest weight of $V^\ast$ here is now $-L_3$. How do I translate this into "expressing" $-L_3$ in terms of $L_1$? I am quite confused as to what I need to show.