In the nLab page, a technique so-called generalized elements is introduced, which is identical to that on MacLane's Categories for the Working Mathematician. We know that in this method, one can check many homological properties of diagrams (such as exactness), but a shortage of this method is that one cannot construct morphisms from this kind of diagram chasing. However, in the very nLab page, it refers to a paper written by Bergman, in which the author claimed that his method of refinement could prevail these deficiencies of MacLane's method: he considers elements (of $X$, say) simply morphisms $A\to X$ for any $A$ (recalling Yoneda's lemma), and instead of considering an equivalence relation generated by refinements, he works directly with refinements: a refinement of $A\to X$ is just another element $B\to X$ which factors as $B\to A\to X$, where $B\to A$ is an epimorphism. It's clear how he uses this to construct morphisms:

Suppose we want to construct a morphism $X\to Y$. Consider a generalized element $\operatorname{id}_X\colon X\to X$ and by diagram chasing, he finds a refinement of $\operatorname{id}_X$, $f\colon A\to X$ which is an epimorphism, with a generalized element $g\colon A\to Y$. In order to construct a morphism $h\colon X\to Y$ which "maps" an "element" $f$ to $g$, namely $h\circ f=g$, we only need to show that the $\ker f\to Y$ induced by $g$ is $0$ (it follows from axioms of abelian categories that $(X,f)$ is the cokernel of $\ker f\to A$). Everything is similar to diagram chasing in abelian groups here.

However, I don't understand how he treats with additions, namely how he "adds" two generalized elements $f\colon A\to X$ and $g\colon B\to X$. As he said, the equivalence relation defined in MacLane's book doesn't respect the abelian group structure of $\operatorname{Hom}(\bullet,\bullet)$. I fail to understand his paragraph on addition (the third paragraph of section 3 of Bergman's paper). Moreover, out of affinity of his method and MacLane's, I cannot imagine how his method escapes from the hell of not-preserving-abelian-structure-of-homs.

In addition, in the nLab page, it mentions that where the actual generalized elements are remembered but a refinement of their domain is allowed, much as familiar from topos theory. I have no idea about the topos theory except that the category of sets is a topos and that topos is something like the category of sets.

Any help is welcome.

Some posts are related: Proving the snake lemma without a diagram chase, https://mathoverflow.net/a/7531/20948

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    $\begingroup$ This comment has nothing to do with the generalized elements, however, if one really cares only about the diagram chasing abilities (in Abelian categories), it seems to me that the more convenient approach is to use the Mitchell's embedding theorem - one starts with a diagram $D$, generates with it the smallest Abelian subcategory $D^{+}$ (here the key thing is that this can always be small), embeds this by the Mitchell's embedding to a category of modules, chases diagrams there and then goes back. $\endgroup$ – Pavel Čoupek Nov 4 '16 at 15:07

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