Prove that the following language is regular
The idea is like the production of two DFAs. Let two DFAs be $M_1 = (Q_1, \Sigma , \delta_1, q_{01}, F_1)$ and $M_2 = (Q_2, \Sigma , \delta_2, q_{02}, F_2)$. Define NFA $M = (Q, \Sigma , \delta, q_0, F)$ to have:

$ Q = Q_1 \times Q_2 \times \{0,1\}$

$ \delta((q_1,q_2,0), a) = (\delta(q_1,a),q_2,1) \forall q_1 \in Q_1, q_2 \in Q_2, a \in \Sigma$

$ \delta((q_1,q_2,1), a) = (q_1,\delta(q_2,a),0) \forall q_1 \in Q_1, q_2 \in Q_2, a \in \Sigma$

$ q_0 = (q_1,q_2,0)$

$ (q_1,q_2,0) \in F \ \text{iff} \ q_1 \in F_1 \ \text{and} \ q_2 \in F_2$ 

Note that the 0 or 1 is just indicating moving turn! In a final state, the turn has to be 0 meaning $M1$'s turn!