I have the following problem. Let $N:L(H_A)\rightarrow L(H_A)$ be a quantum superoperator. The quantum state corresponding to this operator (via Choi-Jamiolkowski Isomorphism) is $\Gamma_A^{N}=id\otimes N(\rho)$, where $\rho$ is density matrix of the maximally entangled state and id is the identity operator. This state is called Choi state. Consider two quantum superoperators $N:L(H_A)\rightarrow L(H_A)$ and $M:L(H_B)\rightarrow L(H_B).$ Show that Choi state of tensor product superoperator $N\otimes M$ is: $$\Gamma^{N\otimes M}_{AB}=\Gamma^{N}_A\otimes \Gamma^{M}_B$$ This problem is taken from Principles of Quantum Communication Theory by Sumeet Khatri and Mark M. Wilde. It is problem 4.6 from this textbook.
Edit. I tried to do this the following way. $$\Gamma^{N\otimes M}_{AB}=\mathbb{I_{AB}}\otimes(N\otimes M)\sum_{iji'j'}|ij\rangle|ij\rangle\langle i'j'|\langle i'j'|=\\ \sum_{iji'j'}|ij\rangle\langle i'j'|\otimes N(|i\rangle\langle i'|)\otimes M(|j\rangle\langle j'|) $$ On the other hand: $$\Gamma^{N}_A\otimes \Gamma^{M}_B=\sum_{ij}\mathbb{I}\otimes N(|ii\rangle \langle jj|)\otimes \sum_{i'j'}\mathbb{I}\otimes M(|i'i'\rangle \langle j'j'|)=\\= \sum_{iji'j'}|i\rangle\langle j|\otimes N(|i\rangle\langle j|)\otimes |i'\rangle\langle j'|\otimes M(|i'\rangle\langle j'|)$$ At this point I tried relabelling but I always encountered the problem that on left-hand side N and M was at 2 and 4 position in tensor product and on right-hand side they are at 3 and 4 position.
