Size of the category of cohomology theories

It is perhaps more natural to study homology theories (or cohomology theories) up to `Bousfield equivalence', where two theories are equivalent if they send the same maps to isomorphisms. (So, for example, classical cohomology with coefficients in a field $k$ are sorted by the characteristic of $k$.) This has been studied, with much sophistication, beginning with papers by Bousfield around 1970, who wrote about the lattice of such localization functors, and gave examples of both orderly and unusual behavior.

I would suggest looking up his papers, and also those of Mark Hovey (e.g. Cohomological Bousfield classes. J. Pure Appl. Algebra 103 (1995), no. 1, 45–59, or Hovey, Mark; Palmieri, John H. The structure of the Bousfield lattice. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 175–196, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.) A paper that shows that versions of your question can be proved to be undecidable is Casacuberta, Carles; Scevenels, Dirk; Smith, Jeffrey H. Implications of large-cardinal principles in homotopical localization. Adv. Math. 197 (2005), no. 1, 120–139. Have fun exploring this!

[I should add that, when restricted to the category of finite CW complexes, the Bousfield classes are known: they are detected by the sequence of Morava K-theories associated to each ordinary prime.]

Welcome to MathOverflow nms! I guess that there are at least two possible answers to your question:

• in algebraic topology, Brown's representability theorem says that cohomology theories are represented by spectra, so you can construct a new cohomology theory by constructing a new spectrum. The category of spectra can also be proved (see Remark 1.4.2.4 in Lurie's "Higher Algebra" for instance) to be presentable, which is a suitable "smallness" condition (see Chapter Five of Lurie's "Higher Topos Theory");
• in algebraic geometry, there are also suitable analogues of Brown's representability, which assert that cohomology theories with suitable properties are representable as motivic spectra, by which one means objects in the categories $mathrm{DA}_tau(S;Lambda)$ and $mathrm{SH}_tau(S;Lambda)$ built out of a scheme $S$, a ring $Lambda$ and a Grothendieck topology $tau$ using $mathbb{A}^1$-homotopy theory (see Ayoub's thesis or the book of Cisinski and Déglise for a detailed analysis of the construction of these categories). Such representability results appear for instance in the paper "Mixed Weil cohomologies" by Cisinski and Déglise, or in §1 of the paper "The rigid syntomic ring spectrum" by Déglise and Mazzari, or in Drew's thesis. I suggest also taking a look at the appendix of the book "Many variations of Mahler Measures" by Brunault and Zudilin, for an application of this representability theorems to Deligne-Beilinson cohomology. Finally, questions about the sizes of the sets of morphisms in the categories $mathrm{DA}_tau(S;Lambda)$ and $mathrm{SH}_tau(S;Lambda)$ are very difficult to answer: for instance, it seems completely out of reach to prove that the motivic cohomology groups $mathrm{Hom}(mathbf{1}_X,mathbf{1}_X(j)[i])$ are finitely generated, under suitable assumptions on $X$ (e.g. $X$ smooth and proper over $mathbb{Q}$).