A monomorphism $m$ in a category is called extremal if the following holds: Whenever $m=f\circ e$ , where $e$ is an epimorphism, then $e$ is an isomorphism.

Dual notion: An epimorphism $e$ is called extremal if the following holds: Whenever $e=m\circ f$ , with $m$ a monomorphism, then $m$ is an isomorphism.