So, I would say that there is a canonical isomorphism between them but that they are not quite the same. So, to take a step back, I'd say that a covector is a

So, I would say that there is a canonical isomorphism between them but that they are not quite the same. So, to take a step back, I'd say that a covector is a linear function from a vector input to a scalar output (those terms themselves being somewhat fungible, the space of "vectors" needs to probably form a module over the "scalars" which must themselves be a ring—the issue is that I would as a physicist include scalar fields but you can have two nonzero fields multiply to give a zero field, so they do not obey the field axioms). Then I would say that an [m n]-tensor is technically a multilinear function from m covectors and n vectors to a scalar. This means that a [0 1]-tensor is really exactly a covector, but a [1 0]-tensor is not really exactly a vector. One can manifestly be constructed from any vector v (take any covector c and produce the scalar c(v)) and we generally have some criteria that I forget which ensures that this is not just an injection but also a bijection implying some inverse, so that these [1 0] "co-covectors" are indeed isomorphic to the vectors. And then of course this becomes much easier in a metric space where we identify a canonical bijection between vectors and covectors called the "metric", this gives a much more direct way by which co-covectors are isomorphic to vectors. Hopefully that all makes sense. The situations in which this is not true are non-metric spaces where linear maps from (linear maps from vectors to scalars) to scalars are just a subset of the "apply to this vector" instances, but other linear maps are maybe also included. Then there is a more dramatic statement which I think requires the space to be Hausdorff-paracompact or something, which is the idea that of course we have the outer product which naturally combines an [a b]-tensor and a [c d] tensor into an [a+c, b+d]-tensor ("take these vectors and covectors, give them to the first tensor, take those vectors and covectors, give them to the second tensor, then take the two scalar outputs and multiply them together), and we claim that actually an [m n]-tensor can always be realized as a finite sum of outer products of m [1 0]-tensors with n [0 1]-tensors, which allows you to do arbitrary index contractions in a geometric way.