Lifting homomorphism when module is direct summand of free module

Question

Let $N,M,M',N'$ be modules such that $F = N \oplus N'$ is a free module.

Let further $f: M \rightarrow M''$ be a surjective homomorphism.

I would like to show that for any homomorphism

$\phi: N \rightarrow M''$ there is a homomorphism $\psi: N \rightarrow M$

such that $f \circ \psi = \phi$, but cannot see any way to do this.

Answer 1

Hints. Let $p_N:F\to N$ be the projection on $N$. Use a basis of $F$ and find $g:F\to M$ such that $f\circ g=\phi\circ p_N$. Now set $\psi=g_{|N}$, the restriction of $g$ to $N$.