Is this proof correct? 

Proof: Let for all $a$ and $e\_G \in G$ we know that if $f$ is a homomorphism from ${(G,\*)}$ to $(H,o)$ then, 

$f(a\*e\_G)=f(a)=f(a)$ o $f(e\_G)$. Similairly $f(e\_G\*a)=f(a)=f(e\_G)$ o $f(a)$. So we now know the following:

$f(e\_G)$ o $f(a)=f(a)$ o $f(e\_G)=f(a)$

At this point can one state that $f(e\_G)=e\_H$? Or will this proof be sufficient if $f$ is a bijection?

Note that I have gone through all the proofs in this [question](https://math.stackexchange.com/questions/1109810/show-that-fe-g-e-h) but no proof given is similar to the one I have written above.