What lies behind the definitions of split monics and epics?

In sets we have two theorems about functions:

Theorem 1. Let $f$ be a function with nonempty domain. The following are equivalent:

1. $f$ is injective.
2. $f$ can be cancelled on the left.
3. $f$ has a left inverse.

Theorem 2. Let $g$ be a function. Assuming the Axiom of Choice, the following are equivalent:

1. $g$ is surjective.
2. $g$ can be cancelled on the right.
3. $g$ has a right inverse.

(The fact that 1 implies 3 is in fact equivalent to the Axiom of Choice; the equivalence of 1 and 2 does not require the Axom of Choice).

When these notions were generalized in category theory, the generalization focused on the second property of each; the reason being that in many instances, those inverses don't exist. For example, in the category $\mathcal{G}roups$ of all groups, one-to-one functions need not have left inverses nor surjective functions right inverses: those are special situations.

However, those special situations are important, as they provide the existence of one-sided inverses. So we still want a categorical way to identify those situations. And those are precisely the "split" cases of monomorphisms and epimorphisms.

Note that if $f$ has a left inverse, then it is certainly left cancellable (hence a monomorphism); and if $g$ has a right inverse, then it necessarily right cancellable, hence an epimorphism. But the converse does not hold.

The "split" cases are the cases that include condition 3 from those two theorems: split monomorphism means "has a left inverse", and split epimorphism means "has a right inverse".