A function $\phi : G \mapsto H$ is said to be an homomorphism iff
$\forall a,b \in G$, $\phi(ab) = \phi(a)\phi(b)$
Is this a complete definition of a group homomorphism. Do I not need to state one more condition that $\phi(e) =e$, where $e$ is the identity element of groups $G$ and $H$.
You have that $$e = \phi(e) \phi(e)^{-1} = \phi(e e) \phi(e)^{-1}= \phi(e) \phi(e) \phi(e)^{-1} = \phi(e).$$ Therefore one doesn't need that to be part of the axioms. Hope that helped you :)