Let $\mathcal{C}$ be a category, and let $A$, $B$, and $C$ be objects.
Given $A \xrightarrow{f} B \xrightarrow{g} C$ such that:
- $f$ is both epic and monic
- $g$ is epic but not monic
- $gf$ is epic and monic
What can we infer about $\mathcal{C}$?
So far all I've really been able to prove is the obvious: $\mathcal{C}$ is not balanced, for if it were, $f^{-1}$ would exist and be monic, so $gff^{-1} = g$ would be monic, which is a contradiction. More generally, we can infer that $f$ does not admit a section for the same reason.
It is also worth noting that this is in fact possible. For example, given the category consisting of precisely $X$, $Y$, and $Z$, with $Hom(X,Y) = \{x_1, x_2\}$, $Hom(Y,Z) = \{y\}$, and $Hom(X,Z) = \{x_1y\} = \{x_2y\}$ and $End(–) = \{Id_–\}$, $y$ is epic but not monic for the only morphism from $Z$ is $Id_Z$; both $x_1$ and $x_2$ are epic and monic, for the only morphism to $X$ is $Id_X$ and the only non-identity morphism from $Y$ is $y$; and $yx_1 = yx_2$ is epic and monic for similar reasons.
Any pointers will be appreciated!
P.S. I find the idea of using the characterisation of epimorphisms by pushouts (see here) and that of monomorphisms by pullbacks (dually) very enticing, as it sets up the potential for some very nice diagram chasing, but despite various pages of increasingly complicated diagrams, nothing has come of it.
EDIT: I should mention that I'm also interested in things we cannot conclude about $\mathcal{C}$ (e.g. we cannot conclude that $\mathcal{C}$ is not complete or not cocomplete, by Eric Wofsey's example in the comments). Of course this is a much more general question and much harder to answer, so I'm not going to properly ask it. That said, any interesting examples (like $CRing^{op}$ below) would be more than welcome, especially in the comments.
