Dear Professor Lawvere, Thanks for your clarifications and views in response to my latest note. Coming from an applications-oriented environment, I do assume a set of Zermelo-Fraenkel axioms with a universe of small sets (as prescribed in CWM) in order to ensure access to a fully viable arithmetic of natural transformations. This seems to allow for more than enough categories for my purposes, but it certainly does give the category of small functions a prominence which can feel artificially restrictive at times. Thus I would be especially attentive to any comments which you might make specifically on the functorial isomorphism (I presume to call it a "Lawvere isomorphism" ) which, in converting the Yoneda picture (function-valued natural transformations) of categorical duality into the Lawvere picture (cocompatible functors), represses the category of small functions and, as I do realize, moves things into the context of the general existence theory of adjunctions and Kan extensions, possibly providing a functorial interpretation of your explanation of the origin of comma categories. By now this isomorphism seems to me to be more of a perspicuous relabelling than a redefiner of concepts, so that I have to plead innocent to your apparent conviction that I agonize over the definition of elements. I am in full accord with the doctrine of elements as you have described it, and the Lawvere isomorphism actually relieves some conceptual agony in this regard by smoothly ensuring that, to within a label, the elements of a function-valued functor constitute a (limit) object which is in the functor's codomain category. But I have to restate my belief that the otherwise perfectly redeemable sentence, "An element of a functor is an attaching functor into the category of elements of the functor," is unacceptably confusing due to the fact that the category of elements of a functor does not in any sense consist of the elements of the functor (as you would describe them). So I would rename it. Pat Donaly