terminology – terminological conflict in the tensor calculation

In the wiki article (I know, not always the source MORE OF CONFIDENCE, and I'm not sure how to add the link) for the covariance and the contravariance of the vectors, almost at the beginning a paragraph starts talking about James Joseph Sylvester and talking about how he named them by the properties of the functors. Then say:

unfortunately, it is the lower index objects (covectors) that generally have setbacks, which are contravariant, while the higher index objects (vectors) have forward thrust, which are covariant. This terminological conflict can be avoided by calling "co-factor" contravariant functors, according to the terminology "covector", and continuing the tradition of treating vectors as the concept and co-factors such as
coconcept

The linked wikis did not help exactly to clarify. For some reason, this originally made sense (I think because I assumed that the vectors had lower indices but I do not remember if I thought there would be a reason for that) and somehow I do not have it anymore now that I've thought more carefully and can show that the covariant basis is covariant, etc. However, the article says that this is the opposite, that the vectors are covariant and the covectors are contravariant, but I can not see why. However, I can see how this thing with the "co" can be a bit confusing. How did this terminological conflict arise and what can be understood by calling co-actors contravariant functors to avoid it? I never (yet) heard those terms used. Thanks in advance for any answer 🙂