In Nielsen and Chuang's QCQI, there is a proof states that
Theorem 8.1: The map $\mathcal{E}$ satisfies axioms A1, A2 and A3 if and only if $$ \mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger} $$ for some set of operators $\left\{E_{i}\right\}$ which map the input Hilbert space to the output Hilbert space, and $\sum_{i} E_{i}^{\dagger} E_{i} \leq I$
To figure out the proof of it, I find in this paper that what Nielsen wants to do is the so called Choi-Jamiołkowski isomorphism. But I just don't understand why Choi operator in the paper I mentioned (eq.8) can be written as $M=\sum_{j}|K_{j}\rangle\rangle\langle\langle K_{j}|.$, where $|K_j\rangle\rangle\equiv I\otimes K_j^T|\Omega\rangle$, and $|\Omega\rangle$ is the unnormalized maximally entangled states, i.e., $\sum_n|n\rangle|n\rangle$.
The aim of eq.(8) is that we don't know if $\varepsilon$ is CP, i.e., we only know the correspoinding choi operator is positive semidefinite, and we want to prove the channel $\varepsilon$ is CP. And there comes eq.(8) which states that any positive semidefinite choi operator can be written as the form $\sum_j|K_j\rangle\rangle\langle\langle K_j|$.
Edit: Three axioms are(I originally thought they are not very related to the main question, so I didn't show it in the main post.):
A1: First, $\operatorname{tr}[\mathcal{E}(\rho)]$ is the probability that the process represented by $\mathcal{E}$ occurs, when $\rho$ is the initial state. Thus, $0 \leq \operatorname{tr}[\mathcal{E}(\rho)\rfloor \leq 1$ for any state $\rho$
A2: Second, $\mathcal{E}$ is a convex-linear $m a p$ on the set of density matrices, that is, for probabilities $\left\{p_{i}\right\}$, $$ \mathcal{E}\left(\sum_{i} p_{i} \rho_{i}\right)=\sum_{i} p_{i} \mathcal{E}\left(\rho_{i}\right) $$ A3: Third, $\mathcal{E}$ is a completely positive map. That is, if $\mathcal{E}$ maps density operators of system $Q_{1}$ to density operators of system $Q_{2}$, then $\mathcal{E}(A)$ must be positive for any positive operator $A$. Furthermore, if we introduce an extra system $R$ of arbitrary dimensionality, it must be true that $(\mathcal{I} \otimes \mathcal{E})(A)$ is positive for any positive operator $A$ on the combined system $R Q_{1}$, where $\mathcal{I}$ denotes the identity map on system $R$.
And the $M$ in the main post is the positive semidefinite choi operator.