TITLE: A proof of the salamander lemma without Mitchell's embedding theorem?
QUESTION [25 upvotes]: The salamander lemma is a lemma in homological algebra from which a number of theorems quickly drop out, some of the more famous ones include the snake lemma, the five lemma, the sharp 3x3 lemma (generalized nine lemma), etc.  However, the only proof I've ever seen of this lemma is by a diagram chase after reducing to R-mod by using mitchell's embedding theorem.  Is there an elementary proof of this lemma by universal properties in an abelian category (I don't know if we can weaken the requirements past an abelian category)?
If you haven't heard of the salamander lemma, here's the relevant  paper.  
And here's an article on it by our gracious administrator, Anton Geraschenko: Click!
Also, small side question, but does anyone know a good place to find some worked-out diagram-theoretic proofs that don't use mitchell and prove everything by universal property?  It's not that I have anything against doing it that way (it's certainly much faster), but I'd be interested to see some proofs done without it, just working from the axioms and universal properties.  
PLEASE NOTE THE EDIT BELOW
EDIT:  Jonathan Wise posted an edit to his answer where he provided a great proof for the original question (doesn't use any hint of elements!).  I noticed that he's only gotten four votes for the answer, so I figured I'd just bring it to everyone's attention, since I didn't know that he'd even added this answer until yesterday.  The problem is that he put his edit notice in the middle of the text without bolding it, so I missed it entirely (presumably, so did most other people).

REPLY [4 votes]: The first paper on abelian categories is "Exact Categories and Duality" D. A. Buchsbaum. It was published in 1955, two years before Grothendieck's famous Tohoku paper. You can find it on jstor. The section 5 is about "fundamental lemmas" such as the Nine Lemma (5.5), the Snake lemma (5.8) and the Five Lemma (5.9). The proofs are direct using the definition of an abelian category (called "exact category" by Buchsbaum, this term was used later by Quillen), in particular they use - of course - no elements. Unfortunately I cannot find the salamander lemma, but lots of basics of homological algebra such as homology, derived functors, satellites. Basically Buchsbaum observed that there is nothing special to module categories treated in Eilenberg-Cartan's landmark book on homological algebra.