  The purpose of this paper is to show that the dual notions of elements &
distinctions are the basic analytical concepts needed to unpack and analyze
morphisms, duality, and universal constructions in the Sets, the category of
sets and functions. The analysis extends directly to other concrete categories
(groups, rings, vector spaces, etc.) where the objects are sets with a certain
type of structure and the morphisms are functions that preserve that structure.
Then the elements & distinctions-based definitions can be abstracted in purely
arrow-theoretic way for abstract category theory. In short, the language of
elements & distinctions is the conceptual language in which the category of
sets is written, and abstract category theory gives the abstract arrows version
of those definitions.
