(Note: in the post below, whenever I use the word "composition", I mean "defined composition", where the domain of the second morphism is the codomain of the first one; likewise, every composition expression "$g\;\;{\scriptstyle \circ}\;\;f\;$" implies the assertion dom($g$) = cod($f$).)
Arbib and Manes define an image factorization system for a category K as a pair (E, M) where E and M are classes of morphisms in K satisfying the following four axioms:
IFS1: E and M are both closed under compositions.
IFS2: The members of E are epic and the members of M are monic.
IFS3: All of K's isomorphisms belong to $\mathbf{E}\cap \mathbf{M}$.
IFS4: Every $f:A\to B$ in K can be factored as $f = m\;\;{\scriptstyle \circ}\;\;e$, where, $e\in \mathbf{E}$ and $m\in \mathbf{M}$, such that, for any other e'\in \mathbf{E} and m'\in \mathbf{M} with f = m'\;\;{\scriptstyle \circ}\;\;e', there exists an isomorphism $\psi$ such that e' = \psi\;\;{\scriptstyle \circ}\;\;e and m' = m\;\;{\scriptstyle \circ}\;\;\psi^{-1}.
Then, in a subsequent exercise, they ask for a proof that, in any category K "in which every morphism can be factored as a coequalizer followed by a monomorphism, ... E = coequalizers and M = monomorphisms yields an image factorization system in K".
I can prove that such (E, M) satisfies IFS2 and IFS3, and that M satisfies IFS1, but I can't prove that for any $e_1, e_2 \in \mathbf{E}$, then also $e_2\;\;{\scriptstyle \circ}\;\;e_1 \in \mathbf{E}$ (i.e. IFS1 for E), nor have I managed much progress proving that all $m\;\;{\scriptstyle \circ}\;\;e$ factorizations ($e\in \mathbf{E}, m\in \mathbf{M}$) satisfy the uniqueness-up-to-isomorphism property of IFS4.
Any help with this proof (or a pointer to a book/paper that gives it) would be appreciated.
Thanks!
Edit:
In my original post, I reworded Arbib-Manes' definition of image factorization system and the statement of the exercise in question. Just in case my rewording is not as faithful to the original as I think it is, I copy the original passages verbatim below.
[p. 38]
13 DEFINITION: An image factorization system for a category K consists of a pair (E, M) where E and M are classes of morphisms in K satisfying the following four axioms:
IFS1: If $e:A\to B\in\mathbf{E}$ and e':B\to C\in\mathbf{E} then e'\cdot e:A\to C\in\mathbf{E}. Dually, if $m:A\to B\in\mathbf{M}$ and m':B\to C\in\mathbf{M} then m'\cdot m:A\to C\in\mathbf{M}.
IFS2: If $e:A\to B\in\mathbf{E}$, $e$ is an epimorphism. Dually, $m:A\to B\in\mathbf{M}$, $m$ is a monomorphism.
IFS3: If $f:A\to B$ is an isomorphism, then $f\in\mathbf{E}$ and $f\in\mathbf{M}$.
IFS4: Every $f:A\to B$ in K has an E-M factorization which is unique up to isomorphism. More precisely, there exists an E-M factorization $(e, m)$ of $f$, meaning $e\in\mathbf{E}, m\in\mathbf{M}$ and $f = m\cdot e$, (so that there exists an object—call it $f(A)$—such that $e$ has the form $e:A\to f(A)$ and $m$ has the form $m:f(A)\to B$), and this factorization is unique in the sense that if (e', m') is another such factorization—$f = m'\cdot e', e'\in\mathbf{E}, m'\in\mathbf{M}$—then there exists an isomorphism [$\psi: f(A)\to C=\mathrm{cod}(e')=\mathrm{dom}(m')$] . . . with \psi\cdot e= e', m'\cdot\psi = m.
. . .
[p. 40]
Exercises
. . .
9 Let K be a category in which every morphism factors as a coequalizer followed by a monomorphism. Prove that E = coequalizers, M = monomorphisms yields an image factorization system in K.