Proof: There is exactly one matrix $B\in K^{n\times m}, AB=I_m$ if $\varphi$ is surjective

Question

Setup: K is a field, $A\in K^{m\times n}$ and $\varphi$ is a homomorphism $\varphi_A :K^n \rightarrow K^m, v\mapsto Av$.

Now I have to show, that there is exactly one matrix $B\in K^{n\times m}, AB=I_m$ if $\varphi$ is surjective.

I have no idea how to start, could you give me a hint on how to start each direction ($\Rightarrow , \Leftarrow$)?