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?