Surjective iff it has a right inverse

Statement

Let $f \colon A \to B$ be a function. Assume the axiom of choice. Then $f$ is surjective if and only if $f$ has a right inverse.

Remarks

If $f \circ h = \mathrm{id}_B$, then each $b$ equals $f(h(b))$, so $f$ is surjective. This direction does not use choice. For the other direction, choose for each $b \in B$ an element $h(b)$ of the nonempty set $f^{-1}(\{b\})$. The statement "each surjection has a right inverse" is equivalent to the axiom of choice.

Explanations

L1 human

For a surjection, each $b$ has at least one input that maps to it. A right inverse picks one such input for each $b$. For infinitely many $b$ with no rule to pick, you need the axiom of choice.