Have a look into Tensor on a vector space, especially at the end
$T^{0,q}(f)=T^{q,0}(f^{*})^{-1}$.
To take
$A^{ijkl}B_{klmn} = \delta^{i}_m \delta^{j}_{n}$ serious you could check out Tensors and Tensors Tutorial as well as Symbolic Tensors to see what it comes down to.
