I'm stuck proving that any regular mono is extremal.
Regular mono: any mono the arises as the equalizer of two morpshisms.
Extremal mono: $f$ extremal mono if $f=f'e$, $e$ epi implies $e$ iso.
I found a reference in "the joy of cats", proposition 7.6.2. The statement is the following:
Let $f: A \to B$ and $g:B \to C$ be morphisms:
1) If $f$ is an extremal mono and $g$ is a regular mono, then $gf$ is extremal.
2) If $gf$ is an extremal mono, then $f$ is an extremal mono.
3) If $gf$ is a regular mono and $g$ is a mono, then $f$ is a regular mono.
The proof of this statement is not such a problem, but I really don't understand how to prove the following
Corollary: any regular mono is extremal.
Hint : Notice that every identity is an extremal monomorphism and apply 1).