Problematic step in proof that Cauchy stress tensor can be expressed as a function of the left Cauchy-Green tensor

I’m studying this proof that Cauchy stress tensor $sigma$ can be expressed as a function of the left Cauchy-Green tensor, i.e.

$$sigma({bf F}) = sigma({bf B}) ;;;;;;;text{where} ;;;;;{bf B} = {bf FF}^T$$

The problem is that the proof starts with the assumption that the deformation gradient $bf F$ transforms according to the rule

$${bf F}^{ast} = {bf FQ}^T$$

where ${bf Q}$ is a rotation tensor, instead of ${bf F}^{ast} = {bf QF}$. Surely I’m missing something.