Clarification of the proof of the Second Isomorphism Theorem

Question

I am reading the proof of the Second Isomorphism Theorem on Dummit and Foote's Abstract Algebra. Could someone please explain how $\varphi$ is surjective?

If $(ab)B$ is any element of $AB/B$, I don't see what element of the domain $A$ is mapped to $(ab)B$.

Answer 1

$(ab)B \sim aB$ since $xB \sim yB$ if there exists some $b \in B$ so that $x=yb$.