What does it mean to take a union of a transition function with a starting state? $$
\delta(q,a) = \begin{cases}
\delta_1(q,a) &amp; q \notin F_1 \text{ or } a \neq \lambda \\
\delta_1(q,a) \cup \{q_1\} &amp; q \in F_1 \text{ and } a = \lambda.
\end{cases}
$$

Look at the second line of the function definition. It's a transition function. $q_1$ is the starting state. What does that union mean here? Union of the function range with $\{q_1\}$?
The function $\delta_1$ is the transition function of an NFA. It outputs a set of states. So $\delta_1(q,a) \cup \{q_1\}$ is just the union of two sets of states.