Holmstrom Cohomological functor

A functor from a triangulated category to an abelian category is called cohomological if for any distinguished triangle (X,Y,Z), the sequence $F(X) \to F(Y) \to F(Z)$ is exact.

See more under Triangulated category. See also the very last paragraph in Hovey’s book, where he seems to define a different (more general???) notion of cohomology functor.

nLab page on Cohomological functor