For a linear (super)-operator $\Psi : \mathbb{C}^{n\times n} \to \mathbb{C}^{m\times m} $, I am wondering whethe
$$ \text{Id}_{k} \otimes \Psi \text{ is positive for each } k\ge 1$$
is equivalent to $$ \Psi \otimes \text{Id}_{k} \text{ is positive for each } k\ge 1$$ In particular, I want to know whether positiveness of $\text{Id}_{k} \otimes \Psi$ is equivalent to positiveness of $\Psi \otimes \text{Id}_{k}$ or not.
Thanks in advance.