nLab
surjection

A function $f$ from $A$ to $B$ is surjective if, given any element $y$ of $B$, $y=f(x)$ for some $x$. A surjective function is also called onto or a surjection; it is the same as an epimorphism in the category of sets.

A bijection is a function that is both surjective and injective.

The axiom of choice states precisely that every surjection in the category of sets has a section. Some authors who doubt the axiom of choice use the term ‘onto’ for a surjection as defined above and reserve ‘surjective’ for the stronger notion of a function with a section (a split epimorphism).