The adapter is likely an AC (wall) to DC adapter. At least this is what we are likely assuming. If so, the answer is no.
Why would you run the adapter to a battery?
You can, however, use a DC to DC converter to drop the 12 V efficiently down to 6V. Lots of these are available on ebay.
I bought a number of these **broken link removed** and they work very well. You can set the current limit and it measures current and is 90% efficient.
It's called a step-down regulator because the source voltage has to be 2 V above the desired voltage.
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This BIGGER typr of question you asked comes up from time to time. An example is you car battery can possibly supply 400 A and the light bulb does not complain. We just don't run wires rated 400 Amps to a 5 W light bulb, but we "protect" the wiring with a fuse.
Your house is another example. You have maybe 200 A at 120 V available per split phase, yet you plug in a 7W night light.