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)
$m$ 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 $g_1 = g_2$.
$e$ 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 $g_1 = g_2$.