It is a bit rough for my taste, but it works.
I am so rusty at this:
Theorem: Let f: A B and g:B C.
Prove that if f and g are onto, then g f is onto.
Since g is onto c C b B such that g(b) = c.
Since f is onto b B a A such that f(a) = b.
Therefore c C a A such that g(f(a)) = c.