In various contexts of topology, algebraic geometry, and algebra (e. g. group representations), one meets the following situation. One has two contravariant functors K and A from a certain category to the category of rings, and a natural transformation p:K-+A of contravariant functors. The Chern character being the central exam­ ple, we call the homomorphisms Px: K(X)-+ A(X) characters. Given f: X-+ Y, we denote the pull-back homomorphisms by and fA: A(Y)-+ A(X). As functors to abelian groups, K and A may also be covariant, with push-forward homomorphisms and fA: A( X)-+ A(Y). Usually these maps do not commute with the character, but there is an element r f E A(X) such that the following diagram is commutative: K(X) | Riemann-Roch Algebra by William Fulton, Paperback | Indigo Chapters