Right inverse ($f\circ g={\mathrm{id}}_Y$)

A right inverse of a function $f:X\to Y$ is a function $g:Y\to X$ satisfying $f\circ g={\mathrm{id}}_Y$. It forces $f$ to be a surjective function. A surjective linear map onto a finite-dimensional vector space has a linear right inverse: lift a basis of the target and extend by linearity.

The assertion that every surjective function has a right inverse is equivalent to the axiom of choice. Choice selects one member of each nonempty fiber of a function, giving a right inverse. Conversely, for a family $(X_i{)}_{i\in I}$ of nonempty sets, the projection from the disjoint union $\{(i,x):i\in I,x\in X_i\}$ onto $I$ is surjective. A right inverse selects one element from each $X_i$, giving a choice function.