Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander Coherent functors. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). Springer, Berlin, pp. 189–231, 1966] and strict polynomial functors in the sense of Friedlander and Suslin (Invent. Math. 127(2), 209–270, 1997). Applications to functor cohomology are discussed. V. Franjou is partially supported by the LMJL—Laboratoire de Mathématiques Jean Leray, CNRS: Université de Nantes, école Centrale de Nantes, and acknowledges the hospitality and support of CRM Barcelona where the final corrections on this paper were implemented.