Interesting to see the discussion of write it up / don´t write it up.... Lets go back to the very basics. If there was any problem:
"Stay with Ac until the Eng arrive"
I´ve lost count of the number of times I´ve encountered the Ac bedded down with the log book open inside with a snag or two that have taken the whole night to trouble shoot. Why all night? Because the crew buggered off and instead of being available for questioning we had to determine in exactly what config the Ac was when the defect appeared.