Of course no, but confronting ideas might be useful to all
Perhaps I got the “supervised” term in the wrong way.
My (wild) guess is that cerebellum will adjust sensory input to cortex expectations (to prevent it to run out of synapses or distinguish seemingly similar sensory inputs in clearly different contexts).
Perhaps an “easier” path to explain myself, is the Dorsal Cochlear Nuclei (DCN). It some sort of cerebellar circuit (Purkinje alike cells) in the auditory pathway. A significant L5 input comes from auditory, visual cortex, and other deeper regions of the cortex (via inferior colliculus). DCN will affect to what from the auditory nerve “passes” to MGN (and hence A1).
My intuition is that cortex in conditioning, via L5 projections, what passes and what no (for example cancel self generated noise, compensate head movement, cancel “echo”, etc…). I think that not only “cancel” but also “amplify” the distinctive part of the auditory nerve.
In the case of the cerebellum, this might be quite more complex, since “sensory” input flow has a feedback (v.gr. sacadic movement, via SC in V1)
“Focus” L5 motor cortex output? Cancel output noise by using other region of the cortex L5. Purkinje are just “frame” interpreters. The whole cortex state is the “frame”. Because of that Cerebellar peduncles are the most fiber dense structure in the whole brain.
Don’t take it seriously