Hi That fitting that "sucks in " is called a taper-lock ..
They are very handy ,especially when the shaft is "just that little bit loose" ,,but ,remember,a machine(tight) fit is allways best .
PS. I once broke a taper-lock insert,being cast-iron,they snap real easy if the fit is sloppy..
Good idea to take your pulley and shaft to the engineers shop so they can advise on the correct taper-lock..
Bruce