# How the superoperator is related with jump operator?

Suppose I have the following Lindblad master equation: $$\frac{d\rho}{dt}=\mathcal{L}[\rho]=-\frac{1}{2}\left(L^\dagger L \rho+\rho L^\dagger L-2L\rho L^\dagger\right)$$ where $$\rho$$ is a single-spin density matrix (i.e., a 2$$\times$$2 matrix), and $$L$$ is the 2$$\times$$2 jump operator, and $$\mathcal{L}$$ is a 4$$\times$$4 superoperator. Now, if I have a two-spin system represented by $$\rho'$$ (i.e., now $$\rho'$$ is a 4$$\times$$4 density matrix) such that $$\frac{d\rho'}{dt}=\mathcal{L}'[\rho']=-\frac{1}{2}\left(L'^\dagger L' \rho'+\rho' L'^\dagger L'-2L'\rho' L'^\dagger\right)$$ where the jump operator still acts on the first spin (i.e., I have $$L'=L\otimes \mathbb{I}$$). How would then the superoperator $$\mathcal{L}'$$ is related to $$\mathcal{L}$$?

The superator $$\mathcal L'$$ would be defined on untangled states $$\rho' = \rho_1 \otimes \rho_2$$ by : $$\mathcal L'[\rho_1 \otimes \rho_2] = \mathcal L[\rho_1]\otimes \rho_2$$ and extend by linearity to a general density matrix $$\rho'$$.