In the slides given by my teacher, the third automaton is the product of the first two:
I tried to do the product myself by doing the transition table of both automata at the same time. I got stuck when I realised that the second automaton has no transition from state $2$ with $a$ as input symbol, thus the automaton should halt at that moment. In the third automaton, I can see there is no transition with input symbol $a$ in any of the states that have $2$ in (sorry for the bad expression, don't know how else to say it).
So when doing the transition table of the two automata, if there is no transition, should I just ignore it like in the 3rd automaton? Or would it be wise to complete the automaton?