An extremal epimorphism (also called a cover) in a category is an epimorphism such that if where is a monomorphism, then is an isomorphism.
The dual notion is an extremal monomorphism.
If has equalizers, then any morphism with the property above must automatically be an epimorphism.
Any strong epimorphism is extremal. The converse is true if has pullbacks.
Any regular epimorphism is strong, and hence extremal. The converse is true if is regular.
An image factorization of a morphism is, by definition, a factorization where is a monomorphism and is an extremal epimorphism.