Alternator and Heatingf Issues

[Mia911]

my battery was not charging so i put the volt meter on it..i got 10.21 v the alternator was giving out, reading on here i see you had to get like 13-14 v output from the alternator, also i thought the car was running a little hot, anyway i decided to take off the alternator to go get it tested, while taking it off i realized the woodruff key was missing off of the fan housing, so that means my fan was spinning faster than it was turning the alternator.....so i still took the alternator to test..as you guys know already it tested out fine...will this of cause the car to run hot since the fan and pulley wasn't spinning the correct speed?...your opinion is greatly considered here guys.....tell me what you think before i put this thing back up......
so i added a woodruffkey to lock the fan/pulley back on the alternator shaft...