Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 18 / 7 / a / Solution

An isomorphism $f:A\to B$ is a morphism with an inverse $f^{-1}:B\to A$ satisfying $f^{-1}f=1_A$ and $f{f}^{-1}=1_B$. A groupoid is a category in which every morphism is an isomorphism.

With the given one-sided inverses,Consequently $f$ is an isomorphism with $f^{-1}=g=h$. This argument uses only the category axioms.