I need to prove that if $L1,L2$ is regular languages $L=\left\{ a_{1}b_{1}a_{2}b_{2}\cdot\cdot\cdot a_{k}b_{k}\mid a_{1}\cdot\cdot\cdot a_{k}\in L_{1},b_{1}\cdot\cdot\cdot b_{k}\in L_{2}\right\} $ will be regular as well.

My approuch is :

There is $A_{1}=\left\langle Q_{1},\Sigma,\delta_{2},q_{0_{1}},F_{1}\right\rangle $ and $A_{2}=\left\langle Q_{2},\Sigma,\delta_{2},q_{0_{2}},F_{2}\right\rangle $

s.t $L\left(A_{1}\right)=L_{1}$ and $L\left(A_{2}\right)=L_{2}$

I was thinking to create a DFA that will recognize $L$

$A=\left\langle \left(Q_{1}\cup Q_{2}\right) ,\Sigma,\delta,q_{0_{1}},F_{2}\right\rangle $

But Im having troubles with the defenistion of $\delta$, how i can force transition from $A_1$ to $A_2$ and vice versa ? Any advices ?