In the wiki article (I know, not alaways the MOST trusted source, and I’m also not sure how to add the link) for covariance and contravariance of vectors, near the beginning a paragraph starts by talking about James Joseph Sylvester and talks about how he named them from properties of functors. It then says:
unfortunately, it is the lower-index objects (covectors) that generically have pullbacks, which are contravariant, while the upper-index objects (vectors) instead have pushforwards, which are covariant. This terminological conflict may be avoided by calling contravariant functors "cofunctors"—in accord with the "covector" terminology, and continuing the tradition of treating vectors as the concept and covectors as the coconcept
The linked wikis didn’t exactly help clarify. For some reason this originally made sense (I think because I assumed vectors had lower indices but I can't remember if I thought there'd be a reason for that) and somehow it doesn’t any more now that I’ve thought of it more attentively and I can show that the covariant basis is covariant, etc. The article however says this is otherwise, that vectors are covariant and covectors are contravariant, but I can't see why. I can see though how this thing with the "co"s could be a little confusing. How did/can this terminological conflict arise and what is meant by calling contravariant functors cofunctors to avoid it? I've never (yet) heard those terms used. Thanks in advance for any answers :)