TITLE: On explaining monomorphisms and epimorphisms
QUESTION [6 upvotes]: I almost always see explanations of monomorphisms such as: if $\forall g_1,g_2, \ m\circ g_1 = m\circ g_2 \Rightarrow g_1=g_2$, then $m$ is monic (and similarly for epimorphisms). I find this is not at all intuitive.
Is it OK to turn the argument around and talk about inequality between arrows instead of equality? (btw, this is for a blog post, not for a scientific paper)
Monomorphisms:
$m$ is monomorphism if does not erase information: if $g_1$ and $g_2$ are different arrows, we can still tell them apart even if they are followed by $m$:
$$
g_1 \neq g_2 \quad \Rightarrow \quad m \circ  g_1 \neq m \circ  g_2\quad (\forall g_1,g_2)
$$
In other words, the fact that $m \circ  g_1 = m \circ  g_2$ can't be because $m$ "hides" the differences between $g_1$ and $g_2$, it really must be because $g_1 = g_2$.
Epimorphisms:
$e$ is an epimorphism if it does not restrict access to information: if $g_1$ and $g_2$ are different, we can still tell them apart if they are preceded by $e$ (e.g. $e$ does not hide the parts of the source where $g_1$ and $g_2$ would have differed):
$$
g_1 \neq g_2 \quad \Rightarrow \quad g_1 \circ  e \neq g_2 \circ  e\quad (\forall g_1,g_2)
$$
In other words, the fact that $g_1\circ e = g_2 \circ  e$ can't be because $e$ "sets things up" so that $g_1$ and $g_2$ look equal, it really must be because $g_1 = g_2$.

REPLY [3 votes]: This is indeed equivalent, as you just take the contrapositive of the usual definition.
I do not agree that this would be better though. We usually use the fact that an arrow is monic or epic to prove that a diagram commutes. The usual definition fits that better. However, this is personal preference and your intuition might be different. Just something you may want to take into consideration.