  We compare derived categories of the category of strict polynomial functors
over a finite field and the category of ordinary endofunctors on the category
of vector spaces. We introduce two intermediate categories: the category of
$\infty$--affine strict polynomial functors and the category of spectra of
strict polynomial functors. They provide a conceptual framework for
compuational theorems of Franjou--Friedlander--Scorichenko--Suslin and clarify
the role of inverting Frobenius morphism in comparing rational and discrete
cohomology.
