Dear list, I am looking for a reference to the following construction. In the simplest case, consider the forgetful functor P:Ring->Set, from the category of commutative rings with a unit. Then the set of natural endomorphisms Nat(P, P) can be identified with the set of polynomials of one variable with integer coefficients Z[X]. This can be easily seen by the chain of isomorphisms Nat(P, P) = Nat(Ring(Z[X], -), P) = Z[X], where the first isomorphism is due to Z[X] being a free ring with one generator and the second is by Yoneda's lemma. Alternatively, just observe that polynomial functions are precisely ones commuting with every ring homomorphism. In general, let P:C->B be a functor from an arbitrary C to a cartesian closed B. Select an object R in C and let R/P:R/C->B be the "obvious" forgetful functor from the co-slice category. For an object S of B define R[S] = Nat(R/P * hom(S, -), R/P), where hom is the internal hom functor of B and * is functor composition in the diagrammatical order. For C = Ring, B = Set this gives the usual ring of polynomials with coefficients in R and variables from S. This construction extends to a functor C x B -> Set and has some nice properties of the usual polynomials: polynomials of "one variable" (i.e., when S = 1) can be composed, to each r:1->R and x:1->S corresponds a polynomial (provided P maps 1 to 1). Has this or dual (where C/R is used instead of R/C) construction been studied? Maybe in enriched contexts? Thank you, Nikita. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]