Invertibility of the canonical morphism ρ in Abelian categories

Construct an explicit inverse for the canonical morphism ρ: Coker(h) → Ker([g]) arising from a composable pair of morphisms h: S → T and g: T → Z in an arbitrary Abelian category, where ρ is induced by the composition Ker(g) → T → Coker(h). Determine whether ρ is an isomorphism under the general axioms of Abelian categories, thereby establishing the bicharadic property without resorting to special embeddings.

Background

In Section 4, the paper investigates bicharadic structures arising from Abelian categories, focusing on the relationship between kernels and cokernels in a composable diagram S → T → Z. A natural morphism ρ: Coker(h) → Ker([g]) is defined via the composition from Ker(g) through T to Coker(h).

For modules over a ring (R-Mod), Proposition 26 shows ρ is an isomorphism, and for small Abelian categories this can be deduced via Mitchell’s embedding into R-Mod. However, in a general Abelian category, the authors do not construct an inverse of ρ and suggest the axioms of Abelian categories might suffice. Resolving this would clarify the general validity of the bicharadic framework beyond module categories.