TITLE: Weil cohomologies with given field of definition and coefficient field
QUESTION [6 upvotes]: Fix a perfect field $k$. Fix a field $K$ of characteristic $0$.
A Weil cohomology induces a functor from the category of smooth projective geometrically connected $k$-schemes to the category of $\mathbb{Z}_{\geq 0}$-graded finite-dimensional commutative associative unital $K$-algebras.
For any two Weil cohomologies are the corresponding functors naturally equivalent?

REPLY [8 votes]: If $k$ is a number field, each embedding $\sigma:k\hookrightarrow\mathbb{C}$ determines a Weil cohomology theory $H^*_{B,\sigma}$ on smooth projective $k$-varieties given by taking the topological cohomology of the complex manifold obtained form $X$ via $\sigma$. In [1], Charles gives an example of $k$, $X$, and two complex embeddings $\sigma_1$, $\sigma_2$ such that $H^*_{B,\sigma_1}(X)$ and $H^*_{B,\sigma_2}(X)$ are not isomorphic as algebras with real coefficients. This gives a counterexample to your question with $K=\mathbb{R}$, since the functors $H^*_{B,\sigma_1}$ and $H^*_{B,\sigma_2}$ are not even pointwise-isomorphic.
[1] Charles, François, Conjugate varieties with distinct real cohomology algebras, J. Reine Angew. Math. 630, 125-139 (2009). ZBL1222.14122.