I saw this question on the internet and found many solutions actually but none of them really persuade me that much.
Question: Given language $L$ which is accepted by a Turing machine $M$, provide a 2-PDA that describes $L$. End of the question.
Note: A Turing machine is given by $M=\{Q,\Sigma,\Gamma,\delta,q_0,q_{accept},q_{reject}\}$.
I assume that the transitions in a 2-PDA look like this:
$$\sigma, \begin{pmatrix} \gamma_1\rightarrow\gamma_2 \\ \gamma_2\rightarrow\gamma_3 \end{pmatrix}$$
Meaning when we see a letter $\sigma$ we remove $\gamma_1$ from the first stack, insert $\gamma_2$ to it, and remove $\gamma_2$ from the second stack and insert $\gamma_3$ to it.
I think there is a simple answer that for a transition $(\sigma_1\rightarrow\sigma_2,L/R)$ in $M$ we can replace it with a transition of the form of a 2-PDA transition but I don't really recognize what is the desired transition.