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In Weibel (Introduction to Homological Algebra)'s proof that left derived functors form a homological $\delta$-functor (Thm. 4.2.6), he does a lot of work that seems unnecessary to me. The relevant pages can be seen on Google Books:

http://books.google.com/books?id=flm-dBXfZ_gC&lpg=PP1&pg=PA45#v=onepage&q&f=false

Once he's established the SES $0\to F(P')\to F(P)\to F(P'')\to0$, aren't the $\partial$s automatically natural (Prop. 1.3.4)?

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    Natural in what sense? They depend on the short exact sequence of resolutions a priori. You now need to refine the comparison theorem of resolutions and the horseshoe lemma to show that you can build a map of these short exact sequences of resolutions and that this map is unique, and that's precisely what Weibel does.2011-07-22

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