Use NFA to express the acceptance of a rule
$\Sigma$ is letter set ${a, b}$. for word $w \in \Sigma^{}$ and 2 language $L_{a}, L_{b} \subseteq \Sigma^{}$ defined on $\Sigma$, we define language $w\left{a \mapsto L_{a}, b \mapsto L_{b}\right} \subseteq \Sigma^{*}$ as follows, by induction on $w$.
$$ \begin{aligned} &\epsilon\left{a \mapsto L_{a}, b \mapsto L_{b}\right}={\epsilon} \ &(a w)\left{a \mapsto L_{a}, b \mapsto L_{b}\right}=\left{w_{1} w_{2} \mid w_{1} \in L_{a}, w_{2} \in w\left{a \mapsto L_{a}, b \mapsto L_{b}\right}\right} \ &(b w)\left{a \mapsto L_{a}, b \mapsto L_{b}\right}=\left{w_{1} w_{2} \mid w_{1} \in L_{b}, w_{2} \in w\left{a \mapsto L_{a}, b \mapsto L_{b}\right}\right} \end{aligned} $$
here, $\epsilon$ represent empty word. for example， if $w=a b a, L_{a}=\left{b^{n} \mid n \geq 0\right}$, and $L_{b}=\left{a^{n} \mid n \geq\right.$ $0}$, then $w\left{a \mapsto L_{a}, b \mapsto L_{b}\right}=\left{b^{\ell} a^{m} b^{n} \mid \ell, m, n \geq 0\right}$.
Furthermore, for languages $L, L_{a}, L_{b} \subseteq \Sigma^{*}$, we define $L\left{a \mapsto L_{a}, b \mapsto L_{b}\right}$ as $\bigcup_{w \in L} w\left{a \mapsto L_{a}, b \mapsto L_{b}\right}$.
For example, if $L=\left{a^{n} b \mid n \geq 0\right}$, $L_{a}={a b}$, and $L_{b}=\left{a^{n} \mid n \geq 0\right}$, then $L\left{a \mapsto L_{a}, b \mapsto L_{b}\right}=\left{(a b)^{m} a^{n} \mid m, n \geq 0\right}$.
Let $\mathcal{A}{0}=\left(Q{0}, \Sigma, \delta_{0}, q_{0,0}, F_{0}\right), \mathcal{A}{1}=\left(Q{1}, \Sigma, \delta_{1}, q_{1,0}, F_{1}\right)$, and $\mathcal{A}{2}=\left(Q{2}, \Sigma, \delta_{2}, q_{2,0}, F_{2}\right)$ be dfa, and for each $i \in{0,1,2}$, let $L_{i}$ be the language accepted by $\mathcal{A}{i}$. 677
Here, $Q{i}, \delta_{i}, q_{i, 0}$, and $F_{i}$ are the set of states, the transition function, the initial state, and the set of final states of $\mathcal{A}{i}(i \in{0,1,2})$, respectively. Assume that the transition functions $\delta{i} \in Q_{i} \times \Sigma \rightarrow Q_{i}(i \in{0,1,2})$ are total. How to give a NFA that accepts $L_{0}\left{a \mapsto L_{1}, b \mapsto L_{2}\right}$? 677