I see that Takuo and I have still not overcome the "royalist" forces. Claudio, Vaughan, Ross and others have mentioned various more "sophisticated" versions of the Yoneda Lemma. However, I contend that the more sophisticated the result is, the LESS it deserves to be called "the fundamental theorem of category theory". I particularly like Euclid's algorithm for the highest common factor as a historical and methodological example, Without meaning to dictate to number theorists how their subject should be organised, let me suggest for the sake of argument that it is a pretty good candidate for being called the "fundamental theorem of number theory". Gauss used the same idea to factorise polynomials, and no doubt number theorists have many more "sophisticated" developments of it. But the principal idea was Euclid's (or one of his colleagues), not Gauss's, and definitely not that of any subsequent number theorist! Saunders Mac Lane said something about "the right" generality, as opposed to the greatest generality. As I say, I like Euclid's algorithm because the idea has survived many many revolutions in the "official" foundations of mathematics. Theorems, like royalty, are rightly the victims of revolutions, because the way in which we encapsulate a piece of theory as a "theorem" depends as much on our current cultural prejudices as it does on the real underlying mathematics. For example, there is Cantor's "theorem" about a powerset being strictly bigger than a set. This belongs entirely to the dogma of set theory. When set theory is overturned, this miserable and wholely misguided "theorem" will go in the dustbin of mathematical history with it. But Euclid's algorithm will live forever. And the Yoneda Lemma will survive as long as Category Theory does in a recognisable form. But it still shouldn't be called the "fundamental theorem"! Paul [For admin and other information see: http://www.mta.ca/~cat-dist/ ]