I'm an undergrad getting familiar with some notions of commutative algebra by reading Atiyah-McDonald. On the exact sequences part, a part of the proof of (2.9) is proving that if $\hom(M'',N)\xrightarrow{\overline{v}}$ is injective for all $N$ (where $\overline{v}(f)=f\circ v)$ then $v$ is surjective.
Any idea on how to prove this? Atiyah makes it seem obvious, I don't see why.