Nason Yanofsky asked a question about composition of dinatural transformations, and there have been a number of replies - but none giving the reference I should have expected, namely [S.Eilenberg and G.M.Kelly, A generalization of the functorial calculus,, J.Algebra 3 (1966), 366 - 375] . What Sammy and I were concerned with were such families as the evaluationn e_A,B : [A,B] o A --> B , where o is a tensor product and [ , ] is an internal hom. Here e_A,B is natural in B in the usual sense; it is also natural, in our extended sense, in the variable A, which appears twice on one side of the arrow, but with two opposite variances. Similar for d_A,B : A --> [B, AoB]. In certain circumstances one can compose such "natural transformations" to form new ones:one example is the composite AoB ---------> [B, AoB] o B ----------> AoB d_A,B o B e_B, AoB which is in fact the identity natural transformation. Sammy and I gave a general treatment in the article above. Later, others generalised our "extended naturals" to get the notion of dinatural transformation. Since these do not compose except in some very special cases, I find them to be of limited interest. In contrast, I find that I still use the Eilenberg-Kellycalculus from time to time. Max Kelly.