I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category.
So far I have the commutative diagram below, with all rows exact:
We examine the homologies
Where $\bar H$ is the dual object of the homology, which is isomorphic to it.
Now the author of the book I'm following (Osborne's Basic Homological Algebra, p.231) says that the connecting morphism is defined as the homology map induced by $f_2$: $$(f_2)_\ast =\delta:\bar H\rightarrow H$$ I do not follow. How do we know such an arrow exists? In what precise sense is it induced by $f_2$?
Added: I would like a detailed explanation; "Just apply the universal properties" is not the answer I'm looking for.