Let $T$ be a torus and suppose $f: T\#T \to T\#T$ is a continuous map. Show that if the induced homomorphism $f_*:\pi_1(T\#T,x_0) \to \pi_1(T\#T,f(x_0))$ is 1-1 then it is onto.
I know that $ \pi_1(T\#T,x_0)=\Bbb Z^2 \star \Bbb Z^2$. Initially, I was thinking that any 1-1 homomorphism would be onto here in $\Bbb Z^2 \star \Bbb Z^2$ but it is not true and I can't see any picture. so please help me here.