Well, the same wiki article has thisI understand that as referring to a single ##y##. That is clearly what is described in the SE link.
Note that the Wikipedia definition of Surjective says “if for every ...” instead of “if for any...”
Does it refer to a single function or all functions?Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.