Sets of morphisms that behave "like" epimorphisms/monomorphisms in a balanced category.

Question

Question. Let $\mathbf{C}$ denote a category, and suppose $E$ and $M$ are wide subcategories such that:

• $g \circ f \in E \rightarrow g \in E$
• $g \circ f \in M \rightarrow f \in M$
• $g \in E \cap M \leftrightarrow g \mbox{ is an isomorphism}$

Does it follow that every $e \in E$ is epic and every $m \in M$ is monic?

Motivation. These conditions ensure that if we're given a factorization into an $E$-morphism followed by an $M$-morphism, then the $E$-morphism is the coimage and $M$-morphism is the image.

Answer 1

Let $\mathcal{C}$ be the category generated by the diagram

_{(source: presheaf.com)}

and the relation $pe=qe$. Let $M$ be the subcategory with morphisms $\{p,q,\mathrm{id}_A,\mathrm{id}_B,\mathrm{id}_C\}$ and let $E$ be the subcategory with morphisms $\{e,\mathrm{id}_A,\mathrm{id}_B,\mathrm{id}_C\}$. $E$ and $M$ satisfy the above conditions, yet $e$ is not epic. You need to impose some saturation conditions to guarantee that morphisms in $E$ are epimorphisms.

On the other hand, as you may already know, all split epimorphisms (resp. split monomorphisms) belong to $E$ (resp. $M$).