In MacLane's 'Category Theory for the working mathematician' there is a definition of exactness (page 200):
'A composable pair of arrows $f: a\rightarrow b$ and $g: b\rightarrow c$ is exact at b if im $f$ $\equiv$ ker $g$ (as subobjects of $b$).'
Here, im $f$ is defined to be ker (coker $f$). In the next paragraph he states that im $f$ $\leq$ ker $g$ if and only if $gf = 0$, and im $f$ $\geq$ ker $g$ if and only if every $k$ with $gk = 0$ factors as $k = m k'$, where $m$ is such that $f = me$ for $m$ monic, $e$ epic.
I cannot figure out why those equivalences hold. Can someone explain how one can prove them?
Note: We are working in an abelian category.
Best Answer
In the meantime I have found a proof for both equivalences, but tiks does not seem to work here, so I will just show you a scan of my handwritten notes.
answer.pdf