The word "stratospheric" means something that is very high up in the atmosphere, like a layer of gas that surrounds the Earth. It can also refer to something that is extremely high in terms of quantity or level. Full definition
The connection does not come as a surprise — the depletion of stratospheric ozone in the Antarctic is already thought to influence the Southern Hemisphere tropospheric climate. (environmentalresearchweb.org)
If you read Raypierre's chapter, you'll see that he was not invoking a grey gas model to require the presence of ozone for stratospheric cooling. (realclimate.org)
This increase in stratospheric temperature with height to the stratopause and the decrease beyond is due to UV solar heating by ozone, independent and unrelated to the greenhouse effect, but nevertheless superimposed on the monotonically decreasing greenhouse temperature profile. (realclimate.org)