In 1968, I was a postdoc at the University of Illinois auditing John Gray's graduate course on categories. I believe John had learned that every functor into sets was a canonical colimit of representables from Gabriel-Zisman: I don't think they give a reference. Coming out of his work on extraordinary naturality with Sammy and on enriched functor categories, Max Kelly had defined (sometime between La Jolla and 1967) the notion of "end" for enriched categories. While browsing in the Illinois Math Dept Library I actually READ Yoneda's paper "On Ext and exact sequences" J Fac Sci U Tokyo 8 (1960) 507-576. I saw that Yoneda had already discovered ends for additive categories, but had not called them that. Yoneda used the integral notation for ends, which I recommended to Brian Day and Max, and they adopted it for their SLNM 106 article. I was learning lots about comma categories from Gray and enjoying translating it into ends to extend the results to enriched categories. Therefore, I claim that Yoneda pre 1960 was well aware of: (i) the end formula nat(f,g) = end(f->g) for the set (or object) of natural transformations; (ii) the coend formula f = coend(A(a,-)*f) expressing each functor into sets (the base) as a canonical colimit of (generalised) representables ("Fourier-like theorem"); (iii) the Lemma which category theorists associate with his name, viz, end(A(a,-)->f) = fa. [I write *, -> for tensor, cotensor, resp.] I also suspect Isbell must have known of the adequacy of the canonical embedding into presheaves in "Adequate subcats" Ill J 4 (1960) 541-552 but have no time now to check. Best regards, _________________________________________________________ / Ross STREET, Professor of Mathematics \ / School of Mathematics, Physics, Computing and Electronics \ / Macquarie University, New South Wales 2109, AUSTRALIA \ / Telephone: 61-2-805-8946 Facsimile: 61-2-805-8241 \ /-----------------------------------------------------------------\ ==============================================================================