Assuming you have these two batteries:
Battery A: 12V, 15 Ah = 180 Wh
Battery B: 5V, 20 Ah = 100 Wh
Can you use battery A to fully charge battery B?
Battery A has less capacity (with respect to charge) compared to battery B, but battery A has more stored energy at 180 Wh.
It seems like there are two ways to think about the transfer (in an ideal case):
Ah: After charging, battery A is left with 0 Ah and battery B is 3/4 charged with 15 Ah.
Wh: After charging, battery A is left with 80 Wh and battery B is fully charged with 100 Wh.
This seems like a situation where, using a water analogy, you have a smaller amount of water (in battery A) trying to fill a larger vessel (battery B). The smaller amount of water is at a higher pressure (and therefore greater energy), but that doesn't make up for the fact that there isn't enough water.
Is this correct? Or is stored energy (i.e. Wh) what really matters and electrons are somehow pulled from the environment to make up for the deficit in stored charge (Ah)?