Relationship between vec operator, Schur, Kronecker and Khatri-Rao product

$\DeclareMathOperator{\diag}{diag}$$\DeclareMathOperator{\vec}{vec}$ Define $\vec(A)$ as the operation of stacking the columns of matrix $A$ into a vector, $A\otimes B$ the Kronecker product, $A\circ B$ the Schur (Hadamard) product and finally $A\diamond B$ Bhatri-Rao product is defined as the column wise Kronecker product.

Here are some useful properties:

$$(A\otimes B)^T = A^T \otimes B^T$$

$$A \diag(x \circ y) B^T = A \diag(x) \diag(y) B^T$$

$$(C\otimes D) ( A \diamond B) = CA \diamond DB$$

$$\vec(AXB^T) = (B \otimes A) \vec(X)$$

$$\vec(A \diag(x) B^T) = (A \diamond B) x$$

$$(A\diamond B \diamond x^T) y = (A \diamond B) (x \circ y)$$

Where $A,B,C,D,X$ are matrices and $x,y$ are vectors of compatible dimensions.