Choi matrices revisited. III

We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send k-superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to k. They also send k-positive maps onto k-block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between k-positivity and Schmidt numbers le k. They also retain the duality between k-superpositivity and k-block-positivity.