Relation between Kraus operators and the Choi matrix

Let $Phi$ be a CPTP map on density operators for a fixed $n-$dimensional state space and fix a basis ${ | jrangle }$. I’m trying to understand the relationship between the Choi matrix $$M_Phi:= sum_{i,j} Phi(|irangle langle j|) otimes |irangle langle j|$$ and a set of Kraus operators ${V_j }$ where $$Phi(rho)=sum_j V_jrho V_j^dagger.$$

The wikipedia page
states without proof or reference that:

The Kraus operators correspond to the square roots of $M_Phi$: For any square root B of $M_Phi$, one can obtain a family of Kraus operators $V_i$ by undoing the Vec operation to each column $b_i$ of $B$.

I have a few questions about this statement.

1. Does anybody have a proof of this statement? It seems strange that a simple rearrangement of the square root of $M_Phi$ gives rise to the Kraus operators.

2. Given a set of Kraus operators ${V_i }$, can one construct the Choi Matrix by reversing the operation described above? That is, from stacking the columns of each $V_i$ to form the columns $b_i= text{Vec}(V_i)$ of a matrix $B$, then would $M_Phi=B^*B$. This seems strange as the Kraus operators $V_j$ can be randomly relabelled and still represent the same channel (i.e. map $V_1 to V_2$ and $V_2 to V_1$). Ultimately this will change the matrix $B$ by some permutation $pi$ which result in a different Choi matrix $M_Phi= B^*pi^* pi B$.