I want to understand why, in the pullback diagram, the map $A \times_C B \to B$ is an epi. I assume $f$ is an epi. I am hoping for a proof from the axioms of an abelian category, so please don't use an embedding theorem unless you also prove it.
Here is my attempt. We want to show $h: A \times_C B \to B$ is epi, so let $d: B \to D$ be an arbitrary epi such that $d \circ h = 0$. Creating the pushout of $(d,g)$ gives us this diagram. I use the result I read about here (which I would love to see a complete proof of) to say that our original square is a pushout. Then, in the diagram here the dotted map is unique. I kept trying stuff after this, but I now think I was reasoning in error, so I'll leave it here.
