covariant contravariant stress tensor

I am currently reading a book about "Material Inhomogeneities in Elasticity" written by Gérard Maugin.

He brings up the covariancy and contravariancy of some well-known stress tensors and in particular that of the Cauchy stress tensor. I was not very familiar with the concept of covariance, so i read up on this very specific tensorial notion. 

I have been able to demonstrate easily that the deformation gradient F is a mixed tensor (covariant and contravariant).