Look at the given definition of monomorphism and try to formulate the definition of epimorphism 
A function $f : A \to B$ is a monomorphism (or monic) if the following holds:
  for all sets $Z$, and all functions $\alpha', \alpha'': Z \to A: f\alpha' =f\alpha'' \implies \alpha' = \alpha''.$

$f: A \to B$ is epimorphism if for all sets $Z$ and for all functions $b: Z \to B$, there's a function $a: Z \to B$ such that  $fa = b$.
I realize this definition looks too much like that of surjection, but I am not sure what else I could do. Please, help me fix it.

The definition you have given is the definition of a split epimorphism.  In the category of sets:


*

*A function is a split epimorphism if and only if is a surjection (one direction is the axiom of choice).

*A function is a surjection if and only if it is an epimorphism.


Therefore, over the category of sets, epimorphisms and split epimorphisms are the same thing.  However, the two definitions are not equivalent in general.  
I don't quite understand the question you've been asked, so I'll just tell you that the correct definition of an epimorphism is exactly dual to the definition of a monomorphism; so you should take the definition of a monomorphism and reverse the arrows and the direction of composition in order to arrive at the definition of an epimorphism.  Then prove that your definition is equivalent to surjectivity.
