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Aug 27, 2021 at 12:49 comment added Mark Grant Hi Peter, there is also an account in Mac Lane's book "Homology", Chapter IV. He doesn't quite discuss naturality but comes close, observing that congruent extensions lead to the same abstract kernel (on p.124).
May 8, 2021 at 11:40 comment added YCor Is this meant to be an answer to your question? If not, the best option is to past this material in your own question (and delete this answer).
May 8, 2021 at 11:31 history answered Peter Kropholler CC BY-SA 4.0