With this question, as with many tutorials of similar questions I’ve found online, my textbook only mentions three currents: $I_1$, which flows through the left loop from and to the 19 V battery, $I_2$, which flows through the right loop from and to the 19 V battery, and $I_3$, which flows through the middle section.
However, why can’t there be an $I_4$ which flows from and to the 12 V battery through the left loop, and an $I_5$ which flows through the right loop? In this diagram, it seems pretty clear that all the current comes from the 19 V battery. But about the current coming from the 12 V battery? Is there no current? (The subsequent analysis completely discounts the presence of any current from the 12 V battery, so I’m thoroughly confused.)