Let $f \colon M \rightarrow N$ be a homomorphism, $K = \ker f$ and $i_k \colon K \rightarrow M$ the inclusion homomorphism.
If $i \colon K' \rightarrow M$ is a monomorphism such that $f i = 0$, then there exists a unique homomorphism $h \colon K' \rightarrow K$ such that $i = i_k h$.
My try:
If $f i = 0$, then $i(K') \subset K$, and then I define $h \colon K' \rightarrow K$ such that $h(k') = i(k')$. Then $$ (i_k h)(k) = i_k(h(k)) = i_k(i(k)) = i(k) \,. $$
Is this right?