Suppose that $n\in \omega$, $m$ is a cardinal and $\kappa, \lambda$ are infinite cardinals. Suppose that any onto function $F:[\kappa]^n \to m$ has the property that there exists a set $H \subseteq \kappa$ with $|H| = \lambda$, such that $F$ is constant on $[H]^n$.
How do I show that if the above is true, then the same is true if I replace $m$ by a smaller cardinal $m'$? I attempted to prove this by taking $F:[\kappa]^n \to m'$ and simply extending the range to $m$. F then becomes a function into $m$ which takes no values in $m \setminus m'$. Then there exists a set $H$ as above and $F$ is constant on $[H]^n$. Then I restricted the range of $F$ back to $m'$.
However this does not work because obviously $F:[\kappa]^n \to m$ is not onto. Does anyone have any ideas as to how to fix this? Thanks very much in advance.
(By $[S]^n$ I mean subsets of size $n$ of $S$).