Here's what I think.
Your smart mug has a heating element in it, as almost all other smartmugs from Target do.
Check their website.
What is the point of knowing the temperature of the liquid in your mug, if you have no way of altering it?
Is that fancy Display to let you know when to drink up or when its to cold.
No.
Also, why would you want the adapter, when 4AAA bats, will last a long time, monitoring level and temp.Unless you wanted to heat something, then they are no longer suitable.
Those batteries are there just like your clock radio needs them to keep time during power failure.
These are my thoughts, so they may be irrelevant to you.
Have fun.