This is a very basic question, but why are dual basis, say, are contravariant objects? Actually I kinda know why but hear me out.
So let $e^i \in V^*$ be dual basis vectors to the basis vectors $e_i \in V$. I understand why basis vectors are covariant because if $(e_k)$ are the underlying basis vectors of $V$, then a new basis say $(f_i)$, must be obtained from
$$f_i = \sum_k A^k_i e_k = A^1_ie_1 + A^2_ie_2 + \dots $$
Now let's assume I claim $f^{p*} = \sum_i B_p^i e^{i*} = B_p^1e^{1*} + B_p^{2*}e^{2*} + \dots$
At this point, I am going to switch into Einstein notation.
Now $$\delta_j^p = f^{p*}(f_j) = B_p^\mu e^{\mu*}(f_j) = B_p^{\mu}e^{\mu*}(A^k_je_k) = B_p^\mu A^k_j e^{\mu*}(e_k) = B_p^\mu A^k_j \delta_k^\mu$$
Set all indices $\mu = k$ and $p = j$ and we get $I = B_p^\mu A_p^\mu$ and $A_p^\mu = (B_p^\mu)^{-1}$. So actually I've turned the basis vectors in $V$ into the real contra variant tensors and the dual basis into covariant tensors. Are these names just conventions?