# Composition of tensor product

I don't have much confidence with density matrices, and I would like to be sure about a property of composition of tensor products operations. Specifically,

$$\sum_i \sum_j |a_i\rangle|b_i\rangle\langle c_j|\langle d_j| =$$
$$=\sum_i \sum_j |a_i\rangle \langle c_j| |b_i\rangle \langle d_j|$$
and the reason for this is the following property of the tensor product: $$(x \otimes y)^T(x \otimes y) = (x^Tx \otimes y^Ty)$$

Am I right?

The simple answer is: yes...