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][1]

Universal cohomologies are already explained here 

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


  [1]: https://www.youtube.com/watch?v=b6IjHwFvcRs&ab_channel=INISeminarRoom1