The present question is a follow up of this question: finite-free-graded modules and the grading of their duals.
Let $k$ be a field, $S=k[x_1,\dots,x_n]$ and $\phi: F \rightarrow G$ a graded homomorphism of graded, finitely-generated, free $S$-modules $F,G$. Denote by $S_+$ the irrelevant ideal of $S$ and suppose that $\phi(F) \subset S_+ G$. Suppose additionally that the maximal degree that appears in $G$ is $m$ and that there are no degrees greater than $m$ in $F$. Now taking the dual of $\phi$ gives $\phi^*: Hom(G,S) \rightarrow Hom(F,S)$. Then the smallest degree that appears in $Hom(G,S)$ is $-m$, let's denote the corresponding component by $S(-m)$.
How can we show that $\phi^*( S(-m)) = 0$?
PS: I can see roughly why this must be the case, but i am looking for a rigorous proof, which is going to further help me understand how the gradings work.