I'm trying to prove that, for positive integers, $m$ and $n$, there is a homomorphism from $F_n$ onto $F_m$ if and only if $m$ is less than or equal to $n$ (where $F_n$ is the universal free group with $n$ generators).
Working on the forwards direction of the proof now -- this is what I have so far:
First, assume that $\phi$ is a homomorphism from $F_n$ onto $F_m$. Assume for the sake of contradiction that $m>n$. Then, for some $x_i$, $\phi(x_i)$ must map to two generating elements in $F_m$, call them $y_a$ and $y_b$.
I get stuck here. I think that because $y_a$ and $y_b$ are non-unique in $F_m$ and part of the definition of a universal free group is that every word is written in a unique way, a contradiction should arise.
Am I on the right track?