  Through abelian categories, homological lemmas for modules admit a self-dual
treatment, where half of the proof of a lemma is sufficient to prove the full
lemma. As remarked in the prequel of the present paper (`Duality in Non-Abelian
Algebra IV'), there is a self-dual treatment of these lemmas even in the case
of non-abelian categories of group-like structures. It is based on the
axiomatic context of a `noetherian form' introduced in the same paper, which is
a generalization of an abelian category that includes among its examples
semi-abelian and Grandis exact categories, as well as many others. The aim of
the present article is to expose the proofs of homological diagram lemmas in
this general context.
