Point out that B can be larger

I think the key point here is that B can/must be larger than A, and that the condition that $g\circ f$ is injective does not constrain us on the part of B that is not touched by f. With this intuition your counter-example comes out of nowhere.

Again not sure how you want to change the solution but I did fix the hint...

You just know how to rewrite hints :)