left inverse

noun2

1. •
   A related function that, given the output of the original function, returns the input that produced that output.

   • In order for a function to have a left inverse it must be injective.
2. •
   For a given morphism f : X → Y, its right inverse (if it has one) is a morphism s : Y → X such that s∘f= mbox id_X.