Let $f,g,h$ be morphisms in a category $C$, for which hold $gf=\operatorname{id}$ and $fh=\operatorname{id}$. Show that $f$ is an isomorphism and $g=h$.
A morphism $f: X\to Y$ in a category $C$ is an isomorphism when there is a morphism $g: Y\to X$ which holds: $gf=\operatorname{id}_X$ and $fg=\operatorname{id}_Y$.
When I can show, that $g=h$, then $f$ is automatically an isormorphism, because it has this property.
It is $g=g\operatorname{id}=g(fh)=(gf)h=\operatorname{id}h=h$