The answer to this question is yes. Roughly speaking: make a category Weil(A) of Weil cohomologies with values in a rigid tensor abelian category A and let Weil(-) be the resulting 2-functor. Then Weil(-) is 2-representable by the universal Weil cohomology. See B. Kahn's lecture

Universal cohomologies are already explained here

https://mathoverflow.net/a/443254/501430

The answer to this question is yes. Roughly speaking: make a category Weil(A) of Weil cohomologies with values in a rigid tensor abelian category A and let Weil(-) be the resulting 2-functor. Then Weil(-) is 2-representable by the universal Weil cohomology. See B. Kahn's lecture Universal Weil cohomology.

Universal cohomologies are already explained here in an answer to What is a cohomology theory (seriously)?.