Here is a product construction for finite automata.
Let $M_1$ = $(Q_1,\Sigma ,\delta_1,q_{01},F_1)$ and $M_2$ = $(Q_2,\Sigma ,\delta_2,q_{02},F_2)$ be two finite automata with the same input alphabet. Let us define $M$ such that $$L(M) = L(M_1) \cap L(M_2) $$
Finite automata for $L(M)$ is defined as $M$ = $(Q,\Sigma ,\delta,q_0,F)$ where $Q$ is a finite set of states, $\Sigma$ is a input alphabet, $q_0$ is a start state, $\delta$ is a transition function and $F$ is final state of a finite automata
- $Q$ = $Q_1$ X $Q_2$ (Cartesian product),
- $\delta((q_1,q_2),a) = (\delta_1(q_1,a),\delta_2(q_2,a))$,
- $q_0 = (q_{01},q_{02})$,
- $(q_{1},q_{2}) \in F $ iff $q_1 \in F_1 $ and $q_2 \in F_2$.
Each state in $Q$ is a pair consisting of a state from $Q_1$ and a state from $Q_2$. For more detail
Reference : https://www.complang.tuwien.ac.at/lkovacs/ATCSNotes/atcs_h2.pdf