How does that have anything to do with what I was saying? The internal energy (per unit mass) of the compressed air at given conditions is not a function of how those conditions were reached.

>The internal energy... is not a function of how those conditions were reached.

How the conditions are reached makes a big difference in efficiency.

I'm baffled at why you focus on minor implementation details like internal energy at a particular snapshot in time, as opposed to highly salient metrics like overall round-trip system efficiency.

Ideally, the round trip efficiency of abiabatic CAES is 100%, as would be the ideal round trip efficiency of isothermal CAES.

What isothermal CAES (with heat storage) would require is a larger storage volume per unit of energy storage capacity, since less heat is generated during the ideal compression. That heat generation is not inefficiency, it's something you want!

Only if the heat is discarded and you reheat with fuel do you benefit from isothermal compression.