Dear Nikita, the special significant case with which you begin, deals with the forgetful functor (which you call P) from rings to sets. It lives in a category (topos) E of covariant functors from rings to sets, and this category you do not give a name. But P and E seem to me to be the main actors in your construction. What you call R[S] (for R in Rings, and S in Sets) is the value at R of the exponent object P^S -> P in E. In particular, R[1] is the value at R of the object P -> P in E. What you observe has as a special case the fact that P -> P, as an object in E, has the universal property of "P[X]", the free P-algebra in one generator; i.e. P[X] = (P -> P) in E. This coincidence, of a colimit type universal property (of P[X]), with a limit type universal property (of the exponential P -> P), is significant, and generalizes to further duality results in E. It also generalizes to any othe algebraic theory T, not just to the theory of rings. For instance, for T the initial algebraic theory (the algebraic theory of sets), it implies that P+1 = (P -> P). For an elaboration of these generalizations, see my "Duality for Generic Algebras", Cahiers 56 (2015), 2-14, or http://home.math.au.dk/kock/DGA03.pdf Anders [For admin and other information see: http://www.mta.ca/~cat-dist/ ]