TITLE: Confused about Weibel proof
QUESTION [6 upvotes]: 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)?

REPLY [3 votes]: I will expand upon Theo's comment a bit.
You are probably confused because Proposition 1.3.4 proves that given a short exact sequence $0\to A\to B\to C\to 0$ of chain complexes there is an associated long exact sequence in homology, and for each map of short exact sequences, there is an associated map of long exact sequences, and in particular the connecting homomorphism is natural in the sense that it commutes with the map of long exact sequences.
The thing is, we don't have a map between the SESs of chain complexes that were used to give the long exact sequence for the derived functors. All we have is the map between the SESs of objects in the Abelian category and we use it to produce a map between the chain complexes $0\to F(P')\to F(P)\to F(P'')\to 0$ and $0\to F(Q')\to F(Q)\to F(Q'')\to 0$, and then that gives the naturality of the connecting map.