I'd like to understand the structure of the functor category Coh whose objects are cohomology functors from a category of Spaces to the category of graded commutative rings GCR. Spaces could be any of the familiar geometric categories: topological spaces, manifolds, algebraic varieties, schemes, etc.
My first question is about just the size of Coh. For any choice of Spaces there are several well known cohomology functors (singular, de Rham, etale, ...) and new ones keep cropping up (syntomic, prismatic) - and they all agree too, on suitable subcategories of Spaces after suitable extensions of scalars, hinting at a fundamental core to it all (e.g., motives) - but I don't know how many more there can be. Is there a systematic way to enumerate or even construct them all? The latter is extremely unlikely because construction of any one we have has been a highly creative and painstaking task, but can we at least know how many in some sense are still out there? Is it even a discrete set, or can we in fact "deform" cohomology theories in families in certain settings?
Same question of size applies to the sets of natural transformations among cohomology theories, i.e., the Hom sets in Coh. Apart from the standard comparison isomorphisms what do we know about other natural transformations, even just for two well known cohomologies, for example, Betti and de Rham?
Sorry if the scope of the question is too broad and I should have made it more manageable by fixing a particular category of spaces and coefficient system for their cohomology. Please feel free to pick a setting that makes for a satisfactory answer.