On Grothendieck transformations in Fulton–MacPherson’s bivariant theory

W. Fulton and R. MacPherson have introduced a notion unifying both covariant and contravariant theories, which they called a Bivariant Theory. A transformation between two bivariant theories is called a Grothendieck transformation  . The Grothendieck transformation induces natural transformations for covariant theories and contravariant theories. In this paper we show some general uniqueness and existence theorems on Grothendieck transformations associated to given natural transformations of covariant theories. Our guiding or typical model is MacPherson’s Chern class transformation c∗:F→H∗c∗:F→H∗. The existence of a corresponding bivariant Chern class γ:F→Hγ:F→H was conjectured by W. Fulton and R. MacPherson, and was proved by J.-P. Brasselet under certain conditions.