I have the next problem. I solved the first item but the second I don't know how to solve the part b).
Let $M,N$ be left $R$-modules and let $f:\longrightarrow N $ be an $R$-Module homomorphism. We define the following sets $\operatorname{Coker}(f)=N/\operatorname{Img}(f), \operatorname{Coim}(f)=M/\ker(f)$. Prove:
a) Let $j: N \longrightarrow \operatorname{Coker}(f)$ the canonical projection. Show that $j \circ f =0$. If $g: N \longrightarrow N'$ is an $R$-module homomorphism such that $g\circ f =0$ exists a unique $R$-module homomorphism $g':\operatorname{Coker}(f) \longrightarrow N'$ such that $g' \circ j=g$.
b) Let $P$ be a left $R$-module and $h: N \longrightarrow P$ be an $R$-module homomorphism such that $h \circ f =0$ and for all $R$-module homomorphism $g: N \longrightarrow N'$ with the condition $g \circ f =0$ there exits $g'': P \longrightarrow N'$ such that $g'' \circ h = g$. Prove that P is isomorphic to $\operatorname{Coker(f)}$.
I think I can use the part a) to prove b) but I dont know how to do it.