  We construct a new Weil cohomology for smooth projective varieties over a
field, universal among Weil cohomologies with values in rigid additive tensor
categories. A similar universal problem for Weil cohomologies with values in
rigid abelian tensor categories also has a solution. We give a variant for Weil
cohomologies satisfying more axioms, like Weak and Hard Lefschetz. As a
consequence, we get a different construction of Andr\'e's category of motives
for motivated correspondences and show that it has a universal property. This
theory extends over suitable bases.
