Dear George, I receive now your question.
The Barr's case (=Exercise VIII.4.6 in Mac Lane's book) and the Snake Lemma seem to have very different canonical connecting morphisms; how does your (beautiful!) general theorem solve this problem?
All these connecting morphisms are canonically induced on subquotients, there is no need of using relations (even though you can, in both cases: a subquotient is the same as a subobject in the cat. of relations, and induced morphisms can always be computed that way: this is already in Mac Lane's Homology.) In the Snake Lemma, with Barr's notation: - Ker h is a subquotient of B (being a subobject of C), - Cok f is a subquotient of B' (being a quotient of A'), - the connecting morphism is induced by g: B --> B'. In the other lemma (Mac Lane, Barr), Ker h and Cok h are both subquotients of the middle object, and the (obvious) connecting morphism is (trivially) induced by the identity of the latter. Subquotients are characterised by a pullback-pushout square with two monos and two epis (in abelian categories; more generally in the Puppe-exact ones; more generally in 'my' homological categories, where you do not have relations). 'Regular' induction just means that there is a commutative cube from the first square to the second. Best wishes Marco [For admin and other information see: http://www.mta.ca/~cat-dist/ ]