An epimorphism $f$ is said to be *extremal*, if for any decomposition $f=i\circ p$ with $i$ a monomorphism, the morphism $i$ is automatically an isomorphism. (This is from the textbook by F.Borceux.) 


Let us say that $f$ is *weakly extremal*, if for any decomposition $f=i\circ p$ with $i$ a monomorphism *and $p$ an epimorphism*, the morphism $i$ is automatically an isomorphism.


Are these definitions equivalent?



Weakened notion of extremal epimorphism?
A counterexample:


Consider the monoid $\langle a,b,c\mid ac=bc\rangle$ as a category with one object. Then $a,b$ are monics, $bc$ is an epic and $c$ is not an epic. So $bc$ is not an extremal epic, but it is easily to see weakly extremal.


