I guess that the point I was trying to make in my post was missed in my telegraphic submission. I'll try to make what I was trying to say clearer. If you take the "representable functor" definition of a product of a object A with an object B, you say that an object P together with an ordered pair of arrows (p_A:P--->A,p_B:P-->B) is a product of A with B if for all objects T , the mapping Hom(T,P)--->Hom(T,A)xHom(T,B), defined by f |---> (p_A.f,p_B.f) is a bijection. This makes it clear that P together with (p_A,p_B) represents the product of A with B and P together with (p_B,p_A) represents the product B with A and that these are not representations of the same functor but that there is a natural isomorphism of such an object with itself which "interchanges the two projections". Note also that this distinction remains even for a product of A with itself which leads a pr_1 and pr_2 notation for the two "projections". Now, as with all such bijections, one can eliminate all reference to sets of morphisms by a "for all x there exists a unique y such that..." statement which here becomes: for all ordered pairs of arrows of the form (x:T--->A,y:T--->B) , there exists a unique arrow f:T--->P, such that (p_A.x,p_B.y)=(x,y), and the distinction made in in set theory between AxB and BxA," that they are not equal, but there is a 'canonical' bijection between them", is perfectly maintained entirely within category theory... or is it? The use of the ordered pair term "(x,y)" still appears in the replacement first order statement, and if the only way that one can use "ordered pair" is through von Neumann's clever but rather grotesque (x,y)={x,{x,y}}, one has to pull in "Peano's entirely spurious singletons" , as Bill Lawvere referred to them. Now if the purpose of an "underlying logic" us to "codify by making explicit the normal habits of reasoning that all mathematicians will accept" and that "an object P and arrows p_A:P--->A and p_B:P--->B" is equivalent to "an object P and arrows p_B:P-->B and p_A:P-->A", or more starkly, the logical equivalence of,"p_A and p_B" and "p_B and p_A". Then there would seem to be no way that a purely first order category theory could make the distinctions that we teach in every elementary math course about the distinction between (x,y) and (y,x), unless we appeal to informal aspects of everyday language where "and" is sometimes commutative and sometimes not, and thus in this case rely on "everybody's having met (x,y) long before they have ever heard of a 'category' ". Apparently this subtlety has surfaced before: In the beginning of Bourbaki's Theorie des Ensembles,\footnote{which, remember, was written by "working mathematicians" who did not consider themselves "professional logicians or professional set-theorists", and may even have had some contempt for them (among others) if we are to judge from a certain fronts piece photograph inserted by Andre Weil into the Fascicule de Resultats.} they introduced as "specific signs", in addition to those of equality and membership, another sign of weight 2, \couple xy, ultimately written as (x,y) together with the Axiom (A3) : (for all x)(for all y)(for all x')(for all y' ) (x,y)=(x',y') implies x=x' and y=y'. They then define "z is an ordered pair (couple)" by "(there exists x)(there exists y)(z=(x,y))", which then gives (x,y) its first and second projections because of the way that "there exists " is constructed (using their "\tau operator"). The existence of the cartesian product of the set A with the set B then follows, as usual, as the set of ordered pairs,since the presence of (x,y) allows them to describe formal "relations" R|x,y| as properties of the ordered pair z=(x,y). Only later do they observe, in an exercise, that the little von Neumann nest of singletons has the property of the axiom A3, but they use this only to show that \couple xy together with A3 is relatively consistent with their other axioms for set theory. It is clear, however, that \couple xy and A3 could have been introduced immediately after they had done quantification and long before any of the axioms for \epsilon were introduced. But, after all, they were trying to use their "Theory of Sets" as a foundation for all of mathematics, so most people have considered this whole business of adding \couple xy and A3 at so fundamental a level an eccentric and superfluous curiosity, and it has all but been completely forgotten. My point is not to pull category theorists back into the intricacies of Bourbaki 's treatment of logic, but rather to point out that the idea of an ordered pair has at least once before been considered a notion that properly belongs somewhere anterior to set theory and can be used in category theory without fear of the latter suffering from any "set-theoretic contamination". In any case, to my eyes, the use of "lists and addresses" with their attendant ordering seems to be pretty fundamental in computer science. Amusingly, Grothendieck "pushed" representability in the forlorn hope that it would convince working mathematicians that they did not have to give up their Cantorian Paradise of " set-theory" in order to make use of the unique insights provided by "category theory", but then had to (re-)invent "universes" when the old paradoxes of set-theory and the category of all sets, all groups, etc. carpingly resurfaced. Bill Lawvere, in contrast, noticed that when the working axioms of set-theory were rephrased in purely "category-theoretic" terms, that they, amazingly, all became "first order" statements , thereby raising the question of an entirely new way to look at foundational questions in which the pesky membership paradoxes could not arise nor even be formally expressible. He, in contrast to Grothendieck, "pushed" the much more radical move of, effectively, "banning all use of Hom-sets" and thereby made the divide crystal clear.

ordering is not inevitable naming is inevitable how do you refer to a proyection withot naming it ? of course naming a proyection by the object it proyects is the mother of all evil have you delt with actions of the symetric group, say, on polynomials on several variables ? There is a related (if not the same) confusion here. what sense has the concept of unlabeled graph ? try to put an unlabeled graph inside a computer ? well, unlabeled graph has to be a quotient by an equivalent relation ... and so on ...