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)?