In Sipser's Introduction to the Theory of Computation, the provided proof for the union operation being closed for regular languages has a step for the transition function that I find a bit lacking.
Assuming that $M_1=(Q_1, \Sigma, \delta_1, q_1, F_1)$ and $M_2=(Q_2, \Sigma, \delta_2, q_2, F_2)$ are two different FSMs for languages $L_1$ and $L_2$. With $M=(Q, \Sigma, \delta, q_0, F)$ being the FSM for $L = L_1 \cup L_2$ (the regular languages for the FSMs). The step for the constructed transition function, $\delta$, states that for each $(r_1,r_2) \in Q$ and each $a \in \Sigma$, let $\delta((r_1,r_2),a)=(\delta_1(r_1,a),\delta_2(r_2,a))$. This is working with the assumption that the input alphabet, $\Sigma$, is the same for $M_1$ and $M_2$.
I am confused on how $\delta((r_1,r_2),a)=(\delta_1(r_1,a),\delta_2(r_2,a))$ behaves for a combined alphabet $\Sigma = \Sigma_1 \cup \Sigma_2$, specifically if $a \in \Sigma_1$ and $a \notin \Sigma_2$? What would the output of $\delta_2(r_2,a)$ be and how would this transition work in the FSMs $M_2$ and $M$?