How can the stress tensor be a contravariant second order tensor?

Question

I just read a proof showing that the stress tensor is a contravariant second order tensor and I cannot reconcile it with what I already knew of second order tensors.

A linear transformation $T\colon V\to V$ is a mixed second order tensor. A bilinear form $B \colon V \times V \to \mathbb{R}$ is a second order covariant tensor. Given that the stress tensor takes a vector and gives back a stress vector I would have expected it to be a mixed tensor like the linear transformation unless somehow its being thought of as a function $V\to V^*$ or $V^* \times V^*\to \mathbb{R}$.

Answer 1

Let me reframe this in other words:

If $V$ is finite dimensional vector space over the field $\Bbb F$ then there is an isomorphism $\hom(V,V)\to V\otimes V^*$.

Here, $\hom(V,V)$ is the vector space of linear transformations $V\to V$, and $V\otimes V^*$ a the space of rank two mixed tensors which correspond to the set of all bilinear maps $V^*\times V\to\Bbb F$.