Monics and epics

Given a category we define a monic $m$ as:

and an epic $e$ as:

where $f$ and $g$ are parallel morphisms. Note that since I don't have fancier LaTeX at my disposal I'm separating the two morphisms $f$ and $g$ by a comma which should be interpreted as two separate morphisms (arrows) between the two objects.

Here's a picture:

To make the idea more familiar let's consider the category $Set$ which is the category of structured sets and total functions between those sets.

If $m$ is injective as a function then it is monic as a morphism (also called a monomorphism).

Similarly if $e$ is surjective as a function then it is epic as a morphism (also called an epimorphism).

There is a one-to-one correspondence between the two relations when considering the category $Set$ and the "real" set but in general this does not hold for arbitrary categories.